fpga control of linear induction motor

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FPGA-based fuzzy sliding-mode control for a linearinduction motor drive

F.-J. Lin, D.-H. Wang and P.-K. Huang

Abstract: A field-programmable gate array (FPGA)-based fuzzy sliding-mode controller, whichcombines both the merits of fuzzy control and sliding-mode control, is proposed to control themover position of a linear induction motor (LIM) drive to compensate the uncertainties includingthe frictional force. First, the dynamic model of an indirect field-oriented LIM drive is derived.Next, a sliding-mode controller with an integral-operation switching surface is designed. Theuncertainties are lumped in the sliding-mode controller, and the upper bound of the lumpeduncertainty is necessary in the design of the sliding-mode controller. However, the upper bound ofthe lumped uncertainty is difficult to obtain in advance in practical applications. Therefore, a fuzzysliding-mode controller is investigated, in which a simple fuzzy inference mechanism is utilised toestimate the upper bound of the lumped uncertainty. With the fuzzy sliding-mode controller, themover of the LIM drive possesses the advantages of a good transient control performance androbustness to uncertainties in the tracking of periodic reference trajectories. A FPGA chip isadopted to implement the indirect field-oriented mechanism and the developed control algorithms

for possible low-cost and high-performance industrial applications. The effectiveness of theproposed control scheme is verified by experimental results.

1 Introduction

A field programmable gate array (FPGA) incorporates thearchitecture of a gate arrays and the programmability of aprogrammable logic device. It consists of thousands of logicgates, some of which are combined together to form a

configurable logic block (CLB) thereby simplifying high-level circuit design. Interconnections between logic gatesusing software are externally defined through SRAM andROM, which will provide flexibility in modifying thedesigned circuit without altering the hardware. Moreover,concurrent operation, simplicity, programmability, a com-paratively low cost and rapid prototyping make it thefavourite choice for prototyping an application specificintegrated circuit (ASIC) [1, 2]. Furthermore, all the internallogic elements and therefore all the control procedures ofthe FPGA are executed continuously and simultaneously.The circuits and algorithms can be developed in the VHSIChardware description language (VHDL) [1, 2]. This method

is as flexible as any software solution. Another importantadvantage of VHDL is that it is technology independent.The same algorithm can be synthesised into any FPGA andeven has a direct path to an ASIC, opening interestingpossibilities in industrial applications in terms of perfor-mance and cost.

The major disadvantage of a FPGA-based system forhardware implementation is the limited capacity of availablecells. Therefore, only research on FPGA-based sliding-

mode or fuzzy controllers can be found in the high-performance control application literature [35]. Chen andTang [3] proposed a FPGA-based sliding-mode controlscheme that used an improved equivalent control methodwithout complicated computations for pulse-width modula-tion (PWM) brushless DC motor drives. A fixed-frequencyquasi-sliding control algorithm based on switching-surfacezero-averaged dynamics is reported in [4]. This FPGA-based system is applied to the control of a buck-basedinverter. Kim [5] proposed the implementation of a fuzzylogic controller to a FPGA system. The fuzzy logiccontroller is partitioned into many temporally independentfunction modules. Then, the FPGA chip is subsequentlyreconfigured one module at a time by using the run-timereconfiguration method. FPGA-based applications ofvarious motor drives can be found in [68]. A digitalwheel-chair controller is presented in [6]. The controlprocess consists in command decoding, speed estimation,and speed serving. Through proper partitioning to con-

current blocks, the design complexity is reduced significantlyfor FPGAs. The concepts of car maneuvers, fuzzy logiccontrol (FLC), and sensor-based behaviours are merged toimplement human-like driving skills by an autonomous car-like mobile robot in [7]. Four kinds of FLCs are synthesisedto accomplish the autonomous fuzzy behaviour control.The implementation of the proposed control on a FPGAchip is addressed. In [8], a sensorless control system forinduction motors, which is realised on a fixed-point digitalsignal processor (DSP) and FPGAs, is presented. Anobserver system was developed to estimate the statevariables of the motor drive.

A linear induction motor (LIM) has many desirableperformance features including a high-starting thrust force,

no need for a gear between the motor and motion devices,the reduction of mechanical losses and the size of motiondevices, a high-speed operation, silence, and so on [9, 10].

The authors are with Department of Electrical Engineering, National DongHwa University, Hualien 974, Taiwan

E-mail: [email protected]

r IEE, 2005

IEE Proceedings online no. 20050205

doi:10.1049/ip-epa:20050205

Paper first received 24th May and in final revised form 15th June 2005

IEE Proc.-Electr. Power Appl., Vol. 152, No. 5, September 2005 1137
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Due to these advantages LIMs have been widely used inindustrial processes and also in transportation applications[11, 12]. The driving principles of a LIM are similar to thoseof a traditional rotary induction motor (RIM), but itscontrol characteristics are more complicated than for aRIM, and the motor parameters are time-varying due tochanges in the operating conditions, such as the speed of themover, temperature and rail configuration. Moreover, thereare significant parameter variations in the reaction railresistivity, the dynamics of the airgap, slip frequency, phaseunbalance, saturation of the magnetising inductance, andend-effects [10, 11]. Therefore, its mathematical model isdifficult to derive completely. Considerable research hasbeen performed on creating models of the dynamicperformance of a LIM which have taken many of thesignificant variations into consideration [913]. However,there still exist uncertainties, including of unpredictableplant parameter variations, external load disturbances,unmodelled and nonlinear dynamics observed in thepractical application of a LIM that need to be considered.Furthermore, since the operation of a LIM involves twocontacting bodies, a frictional force is inevitably among theforces of motion. In addition, this friction characteristicmay be easily varied due to changes in normal forces

in contact, and also the temperature and humidity. In aclosed-loop control system, the frictional force results in asteady-state error, a limit cycle and a low bandwidth[14, 15]. Unfortunately, friction is a natural phenomenonthat is quite difficult to model, and it is not completelyunderstood. Therefore, it is impossible to obtain a precisefriction model for practical applications. However, adynamic model of a LIM can be obtained by modifyingthe dynamic model of a RIM at certain low speeds since aLIM can be visualised as an unrolled RIM. Thus, field-oriented control [13, 16] can be adopted to decouple thedynamics of the thrust force and the flux amplitude of theLIM.

There have been many studies that have investigated therelationship between sliding-mode control and fuzzy controlin an attempt to combine these two techniques. Yager andFilev [17] determined fuzzy rules for sliding-mode condi-tions. Wu and Liu [18] used the sliding modes to determinethe best values for parameters in fuzzy control rules thatwere used to improve the robustness of the fuzzy control.A Lyapunov function and a boundary layer have beenemployed to satisfy the reaching condition and to avoidchattering, respectively. Liu and Lin [19] used a fuzzycontroller to adjust the sliding surface of a sliding-modecontroller. A fuzzy sliding-mode controller is investigated inLin and Chiu [20] in which a simple fuzzy inferencemechanism is used to estimate the upper bound on the

uncertainties for the position control of a permanent-magnet synchronous motor.

The motivation of this study is to design a suitablecontrol scheme to confront the uncertainties that exist in aLIM drive including the frictional force using a FPGA chipto allow possible low-cost and high-performance industrialapplications. Due to its robustness and case of implimenta-tion, a fuzzy sliding-mode position controller is adopted inthis study to control the mover position of an indirect field-oriented control LIM drive. The proposed control algo-rithms are realised on a 24 MHz FPGA (XC2V1000) with a1000 000 gate count and 10 240 flip-flops from Xilinx, Incusing VHDL. The design and implementation of theFPGA-based control IC will be described in detail.

Compared with a DSP or a PC-based fuzzy controller,the merits of the FPGA-based fuzzy controller are a parallelprocessing and small size in addition to a low cost.

Moreover, the developed VHDL code can be easilymodified and implemented to control any type of ACmotors as well.

2 Indirect field-oriented linear induction motordrive

The primary (mover) of the adopted three-phase LIM issimply a cut open and rolled flat rotary-motor primary. Thesecondary usually consists of a sheet conductor usingaluminium with an iron back for the return path of themagnetic flux. The primary and secondary form a single-sided LIM. Moreover, a simple linear encoder is adoptedfor the feedback of the mover position.

The dynamic model of the LIM is modified from thetraditional model of a three-phase, Y-connected inductionmotor in a synchronous rotating reference frame and can bedescribed by the following differential equations [10, 13]:

_iqs p

hveids

Rs

sLs

1 s

sTr

iqs npLmp

sLsLrhvldr

Lm

sLsLrTrlqr

1

sLsVqs 1

_ids Rs

sLs 1 s

sTr

ids phveiqs L

m

sLsLrTrldr

npLmp

sLsLrhvlqr

1

sLsVds 2

_lqr Lm

Triqs

p

hve np

p

hv

ldr

1

Trlqr 3

_ldr Lm

Trids

1

Trldr

p

hve np

p

hv

lqr 4

Fe Kfldriqs lqrids M_v Dv FL 5

where Tr

Lr/R

r, s

1

L2

m=L

sL

m, K

f 3n

ppL

m=2hL

rand Rs is the winding resistance per phase, Rr isthe secondary resistance per phase referred primary, Lm isthe magnetising inductance per phase, Lr is the secondaryinductance per phase, Ls is the primary inductance perphase, ve is the synchronous linear velocity, v is the moverlinear velocity, ldr, lqr are the d-axis and q-axis secondaryflux, ids, iqs are the d-axis and q-axis primary current, Vqs,Vds are the d-axis and q-axis primary voltage, Tr is thesecondary time-constant, s is the leakage coefficient, Feis the electromagnetic force, Kf is the force constant, FL isthe external force disturbance, M is the total mass of themoving element, D is the viscous friction and iron-losscoefficient, h is the pole pitch and np is the number of pole

pairs.In an ideally decoupled induction motor, the secondary

flux linkage axis is forced to align with the d-axis. It followsthat:

lqr 0; _lqr 0 6

Using (6), the desired secondary flux linkage in terms of idscan be found from (4) as:

ldr Lm=Trs 1=Tr

ids 7

where s is the Laplace operator. Moreover, using (3) thefeedforward slip velocity signal can be estimated using ldr

shown in (7) and iqs as follows:

vsl Lm

Trldriqs 8

1138 IEE Proc.-Electr. Power Appl., Vol. 152, No. 5, September 2005
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The block diagram of an indirect field-oriented LIM systemis shown in Fig. 1, where dis the position of the mover; d isthe position command; v is the velocity command; ids is thecommand of flux current; ids is the control effort. Theindirect field-oriented LIM system consists of an LIM, aramp comparison current-controlled PWM voltage sourceinverter (VSI), an indirect field-oriented mechanism, acoordinate translator, cos ye and sin ye generator, where ye isthe position of the secondary flux, a speed control loop, anda position control loop. Three-phase current commands,

i

a; i

b and i

c are generated from the coordinate translatorfor the ramp comparison current controller. The LIM usedin this drive system is a three-phase Y-connected two-pole3 KW 60 Hz 180 V/14.2A type. The detailed parameters ofthe LIM are:

ids 2:35 A; Rs 5:3685O; Rr 3:5315O; h 0:027 m

Lm 0:024 19 H; Ls 0:028 46 H; Lr 0:028 46 H 9

By use of the indirect field-oriented control technique andwith the fact that the electrical time-constant is much

smaller than the mechanical time-constant, the electromag-netic force shown in (5) can be reasonably represented bythe following equations:

Fe KFiqs 10

KF 3

2np

pL2mh Lrids 11

A curve fitting technique based on a step response is appliedto find the drive model off-line at the nominal case (F

L0).

The results are (on a scale of 14.9717 (m/s)/V)

KF 33:73 N=A; M 2:78 kg 41:6213 Ns=V;

D 36:0455 kg=s 539:6624 N=V 12

The F symbol represents the system parameters in thenominal condition. Although the electromagnetic forcecan be simplified as (10) via the field-oriented control,considering the variations of system parameters andexternal nonlinear and time-varying disturbance including

limiter

sin/cos

table

d v

encoder

three-phase

220 V

60 Hz

LC

Ta

Tb

Tc

ia

ib

ia*

d* v* i*qs

cose+ i*

dssin

e

ib* i

c*

position

controller

speed

controller

coordinate

translator

integratorv

sl

np

npv

ve

encoder

signal

encoderinterface

v

d

+

+

+

+

+

e

h

e

generator

Tr

dr

Lm

LIM

ramp

comparison

current

control

rectifierPWM

inverter

sinecos

e

( i*qs

cose+ i*

dssin

e)

12

(i*qs

sine+ i*

dscos

e)3

2

( i*qs

cose+ i*

dssin

e)

12

(i*qs

sine+ i*

dscos

e)3

2+

i*qs

i*ds

e

sine

&cos

egenerator

Fig. 1 System configuration of the LIM drive



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frictional force, the LIM drive system is a nonlinear time-varying system in practical applications.

3 Fuzzy sliding-mode control system

The proposed sliding-mode position controller is shown inFig. 2. The state variables are defined as follows:

x1t d d 13

_x1t _d

_d _d

x2t 14

where x2t _d v. Then a LIM drive system can berepresented in the following state-space form:

_x1t_x2t

!

0 10 D=M

!x1tx2t

!

0

KF=M

!iqt

0

1=M

!FL

1

0

!_d

15

The above equation can be represented as:

_Xt AXt BUt CFL G _d

16

where

A 0 1

0 D=M !

; B0

KF=M !

; C0

1=M !

;

G1

0

!; Ut iqst

Consider (15) with uncertainties:

_Xt A DAXt B DBUt

CFL ffv G _d

17

where DA and DB are denoted as the uncertaintiesintroduced by system parameters M and D. Moreover,ff(v) is the frictional force. Considering Coulomb friction,viscous friction and the Stribeck effect, the frictional force

can be formulated as follows [14, 15]:

ffv FCsgnv FS FCev=vs

2

sgnv Kvv 18

where FC is the Coulomb friction; FS is the static friction; vSis the Stribeck velocity parameter; Kv is the coefficient ofviscous friction; sgn( ) is a sign function. Reformulate (17),then:

_Xt AXt BUt Et 19

where E(t) is call the lumped uncertainty and is defined by

Et BDAXtBDBUt

BCFL ffv BG _d

20

and B+ (BTB)

1BT is the pseudo-inverse. According to(19), an integral-operation switching surface is designeddirectly from the nominal values of system parameters A

and B for the sliding-mode position controller as follows[20]:

St C Xt

Zt0

A BKXtdt

! 0 21

where C is set as a positive constant matrix, and K is astate-feedback gain matrix.

From (21), if the state trajectory of system (19) is trapped

on the switching surface (21), namely St _St 0, thenthe equivalent dynamic of system (19) is governed by the

following equation:

_Xt A BKXt 22

It is obvious that the position error x1(t) will converge tozero exponentially if the pole of system (22) is strategicallylocated on the left-hand plane. Thus, the overshootphenomenon will not occur, and the system dynamic willbehave as a state-feedback control system. Based on thedeveloped switching surface, a switching control law whichsatisfies the hitting condition and guarantees the existence ofthe sliding-mode is then designed. Now a position controlleris proposed in the following [20]:

Ut KXt f sgnSt 23

where sgn( ) is a sign function and f is defined as 7E(t)7rf.In the general sliding-mode control, the upper bound of

the uncertainties, which includes parameter variations andexternal load disturbances, must be available. However, thebound of the uncertainties is difficult to obtain in advancefor practical applications. Therefore, a fuzzy sliding-modecontroller is proposed here, in which a fuzzy inferencemechanism is used to estimate the upper bound of thelumped uncertainty. The fuzzy inference mechanism canconstruct the estimation model of the upper bound ofthe lumped uncertainty. Compared with a conventionalestimator, the fuzzy inference mechanism uses prior

expert knowledge to accomplish the control object moreefficiently.

Replace f by Kf in (23), the following equation can beobtained:

Ut KXt Kf sgnSt 24

where Kf is estimated by a fuzzy inference mechanism.Because the data manipulated in the fuzzy inferencemechanism is based on fuzzy set theory, the associatedfuzzy sets involved in the fuzzy control rules are defined asfollows:

N: negative

NH: negative huge

NS: negative smallPM: positive medium

Z: zero

NB: negative big

Z: zeroPB: positive big

P: positive

NM: negativemedium

PS: positive smallPH: positive huge

Since only three fuzzy sets, N, Z and P, are defined for S

and _S, the fuzzy inference mechanism only contains ninerules. The resulting fuzzy inference rules are as follows[20]:

Rule 1: IF S is P and _S is P THEN Kf is NH

Rule 2: IF S is P and _S is Z THEN Kf is NB

Rule 3: IF S is P and _S is N THEN Kf is NM

Rule 4: IF S is Z and _S is P THEN Kf is NS

Rule 5: IF S is Z and _S is Z THEN Kf is ZE

Rule 6: IF S is Z and _S is N THEN Kf is PS

Rule 7: IF S is N and _S is P THEN Kf is PMRule 8: IF S is N and _S is Z THEN Kf is PB

Rule 9: IF S is N and _S is N THEN Kf is PH

sliding-mode

position controllerd*

i*qs

d v

+

_

fuzzy inference

mechanism

Kf

s

d

S(t)

field-oriented control LIM

S(t)

Fig. 2 Fuzzy sliding-mode control system

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The fuzzy output Kf can be calculated by the centre ofarea defuzzification as:

Kf

P9i1

wici

P9i1

wi

c1 . . . c9

w1

.

.

.

w9

264

375

P9i1

wi

tTW 25

where t c1; . . . ; c9, c1 through c9 are the centre of the

membership functions of Kf; and

Ww1; . . . ;w9 P9i1

wi

is a firing strength vector.

4 Circuit design on a FPGA chip

The block diagram of the FPGA-based control system for aLIM drive using a current-controlled technique is shown inFig. 3. The current-controlled PWM VSI is implemented byan IPM switching component (MUBW30-06A7) manufac-tured by IXYS Co. with a switching frequency of 15 KHz.The timing control module, encoder interface module, thefield-oriented control module and the fuzzy sliding-modecontrol module are realised on the FPGA chip. Three-phasecurrent commands, ia; i

b and i

c are generated from the

coordinate translator and sent to three D/A converters forthe ramp comparison current control. The adopted D/Aconverters are 12 bits in size with an output voltage of75 V. There are a 4-bit control bus and an 8-bit data bus

between the FPGA and D/A converter. Multiplexing isimplemented in the data bus to form the 12 bits of data. The12 bits of data are divided into a least significant 8-bit latchand a most significant 4-bit latch. The 4-bit control buscontrols the loading of data to the input latches of the D/Aconverter. The entire I/O port for this chip includes two pinsfor the input ports and 36 pins for the output port.Moreover, 8281 out of 10 240 flip-flops (80%) have beenused in the FPGA chip.

4.1 Encoder interface moduleA block diagram of the encoder interface is shown inFig. 4a, which consists of the timing control, two digitalfilters, a decoder, an up-down counter, two clock (CLK)generators, a register, a command generator and twoadders. The function of the encoder interface is to obtain theposition and speed values of the mover. The resolutions ofthe encoder are 0.1 m2000 digital value and 1.5915m/s 44digital value at a sampling frequency of 732 Hz. The scaling1 V 14.9717 m/s is obtained since the specification isdesigned as 1 V 409 digital value. The pulse count signalPLS and the rotating direction signal DIR are obtainedusing the A, B pulse input signals from the decoder throughtwo digital filters. The position signal d can be obtainedusing the PLS and DIR signals through the up-downcounter. Moreover, the command generator includesperiodic sinusoidal intellectual property (IP) and periodicstep IP in order to generate the position command d. Thex1 signal results from the difference between the positioncommand d and the position signal d. Furthermore, the x2signal is the velocity signal v, and results from the differencebetween the position signal dand the time delay of d, ddelay.

L

C

+

encoder

signal

encoder

x1

x2

x1

x2

SK

f

v

data & D/A

controller

12 36

2

12

1212

12

12

12

12

12

1212

12

12

LIM

three- phase

220 V

60 Hz

rectifierPWM

inverter

Ta

Tb

Tc

ia

ib

ic*

ic*

i*bi

*b

i*ds

i*qs

ia*

ia*

ramp

comparison

current

control

D/ A

converter

data & D/A

control signal

S

x1

generator

e

generator

command

generator

encoder

interface

encoder interface modulefuzzy sliding-mode control module

sliding-mode

position controller

fuzzy inference

mechanism

integral switching surface

coordinate

translator

24 MHz

CLK

FPGA

timing control module

field-oriented control module

sin(e

)&

cos(

e)

generator

Fig. 3 Block diagram of the FPGA-based control system



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4.2 Field-oriented control moduleThe field-oriented control mechanism shown in Fig. 4b iscomposed of a ye generator, a coordinate translator, a cos yeand sin ye generator and timing control. The ye is obtainedusing the estimated slip velocity signal vsl based on (8), thecontrol effort signal iqs, the velocity signal v and an

integrator. Then, sinye and cosye signals are obtainedthrough the cosye and sin ye generator. Moreover, three-phase current commands, ia; i

b and i

c are generated from

the coordinate translator, which consists of six multipliers

and five adders, and sent to three D/A converters for theramp comparison current controller. Each D/A converterneeds 12 pins in the output port.

4.3 Integral switching surface moduleA block diagram of the integral switching surface is shownin Fig. 4c. First define:

C g1g2

!T; Kk1k2

!T26

a

D Q

d

v

12

12

12

12

12

12

2

d

encoder interface module

+

+d*

x1

generator

encodersignals

timing control

up-downcounterdecoder

command

generator

encoder interface

A pulse

B pulse

24 MHz CLK

digital

filter

digitalfilter

375 kHz CLK

generator

>C

D Q

>C

DIR

PLS

adderx

1

x2

732 Hz CLK

generator

12 dregister

adder

ddelay

>

+

+

+

+

++

+

nv

+

+

+

12

12

12

12

12

121212

12

12

1212

1212 12

12

12

12

12

12

12

v

12

+

12

12

12

12

12

vp

e

12

12

12

12

12

12

12

12

12

12

timing controlfield-oriented control module

e

generator

np

multiplier

multiplieradder

i*qs

i*qs

i*qs

cose

i

*

qs cosei

*

qs cose

i*qs

sine

i*qs

sine i*

qscos

e

i*qs

i*qs

i*ds

Tr

24 MHz

CLK

divider vsl

ve

h

integrator

e

coordinate

translator

sine

& cose

generator

sin/cos

table

sine

sine

cose

cose

12

12

12

12i*ds

cose

12i*qs

sine

12i*ds

sine

i*ds

sine

i*ds

sine

i*ds

sine

i*ds

cose

i*ds

cose

i*a

i*b

i*c

i*ds

i*ds

sine

cose

multiplier

multiplier

multiplier

multiplier

adder

adder

multiplier

12

12 multiplier

1/2

3/2

adder

adder

adder

b

Fig. 4 Circuits designed on the FPGAa Encoder interface module

b Field-oriented control module

c Integral switching surface module

d Sliding-mode position control module

e Membership functions of the fuzzy sets

f Fuzzy inference mechanism module



7/12

For the purpose of scaling (1 V 14.9717 m 409 digitalvalue), x1 should be divided by 732. Then, define a 1/732,and (21) can be rewritten as follows:

St b1x1 g2x2

Zt0

b2x1 g1x2 b3x2dt 27

where b1 ag1, b2 ag2k1KF=Mandb3 g2k2KF D=M.Therefore, five multipliers, four adders, an integrator and a

register are needed to implement (27). Moreover, the _Ssignal can be obtained using the difference between signal Sand its delay Sdelay.

4.4 Sliding-mode position control moduleA block diagram of the sliding-mode position control isshown in Fig. 4d. First, (23) is rewritten as follows:

U b4x1 k2x2 f sgnSt 28

where b4 ak1. Then, to avoid the chattering phenomena,the sign function in the sliding-mode control is replaced bythe following function:

St

Stj j d29

S

+

+

+

+

+

+

+

12

12

12

12

12

12

12

12

12

12

12

12

+

12

12x1

x2

12

12

timing control

integral switching surface module

1

2

x1

2

x1

3

x2

x2

1

x2

Sdelay

multiplier

multiplier

multiplier

multiplier

multiplier

S

adder

adder

adder

register

12

12

12

12

12

12

1

x1

2

x1

3

x2

12

12

2x

2

1

x2

adderintegrator

c

0

f

+

+

++

+

S

+ S

12

12 12

12

>

0

+

+

Kf

timing control

sliding-mode position control module

i*qs

x1

x2

12

x2

12

12

S

12

S

for Kf

for Kf

k2

1212

12

12

12

12

12

12

12

12

1212

12

12

12

x1

4

absolute

value

multiplier

multiplier

multiplier

adder

adder

comparator

multiplexer

24 MHz

CLK

divider

adder

4x

1+ k

2x

2

++ S(t)

S(t)

++ S(t)

S(t)

d

or

Fig. 4 Continued



8/12

where

d 0; Stj j ! Zd0; Stj joZ

&

and d0 and Z are positive constants. Moreover, the controleffort iqst can be obtained using (28) and (29).

4.5 Fuzzy inference mechanism module

The membership functions for the fuzzy sets correspondingto the switching surface S, its derivative _Sand upper boundof the lumped uncertainty Kf are defined in Fig. 4e where

mS, m _S and mKf are the respective membership grades. The

membership functions are designed as a symmetricallytriangular form and c1 through c9 are the centre of themembership functions of Kf. Their universe of discoursesare all assigned to be [2047, 2047] and the membershipgrades of mS and m _S are assigned as [0, 2047]. A blockdiagram of the fuzzy inference mechanism is shown inFig. 4f including the timing control, fuzzification, firing

strength calculation and defuzzification circuits. Themembership grades of the membership functions of S and_S can be obtained using the fuzzification circuit. The vN1,

e

P

S,S

NH ZE PSNB NM NS PM PB PH

N

Kf

S

,

S

Kf

2047Z

02047 2047

02047 2047

c1

c2

c3

c4

c5

c6

c7

c8

c9

f

+

min

min

min

min

min

min

min

min

+

+

+

+

++++++++

+

+

+

+

+

>

timing control

fuzzy inference mechanism module

defuzzification

firing strength calculation

vP1 12

vN1

vZ1

vP1

vN2

vP2

vZ2

12

12

12

12

12

12

w1

w1

12

12

c1 12 12

12

12

12

12 12

1212

12

12

12

12

c2 12

c3 12

c4 12

c5 12

c6 12

c7 12

c8 12

c9 12

w2 12

w3 12

w4 12

w5 12

w6 12

w7 12

w8 12

w9 12

w1 12

w2 12

w3 12

w4 12

w5 12

w6 12

w7 12

w8 12

w9 12

w212

w312

w412

w512

w612

w712

w812

w912

vP1 12

vP1 12

vN2 12

vN1 12

vN1 12

vN1 12

vN2 12

vN2 12

12

vP2

vZ2

12v

Z2

12v

Z2

12v

Z1

12v

Z1

12vZ1

12

vP2 12

vP2 12

min

33 rule table

fuzzification

12 S

12 S

multiplier

multiplier

multiplier

multiplier

multiplier

multiplier

multiplier

multiplier

multiplier

adder

adder divider

24 MHz

CLK

Kf

Fig. 4 Continued



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vZ1 and vP1 represent the membership grades of the fuzzysets N, Z and P of S, respectively. Moreover, vN2, vZ2 andvP2 represent the membership grades of the fuzzy sets N, Z

and Pof _S, respectively. Furthermore, the firing strengths ofthe linguistic rules w1 to w9 are obtained according to themin operator, which is mainly consists of the comparator.In addition, Kf is resulted using the defuzzification circuitwith w1 to w9 as the inputs. The defuzzification method isbased on (25). Therefore, the defuzzification circuit includesnine multipliers, two adders and one divider.

5 Experimental results

The parameters of the sliding-mode controller are given asfollows:

k1 900; k2 180; g1 2; g2 2 30

These parameters are chosen to achieve the best transientcontrol performance in the experimentation considering therequirement of stability and possible operating conditions.The control objective is to control the mover to moveperiodically according to reference trajectories. Two testconditions are provided, which are the nominal conditionand the parameter variation condition. In the experimenta-tion, the parameter variation condition is the addition ofone iron disk with a 8.34 kg weight to the mass of themover.

Some experimental results are provided to demonstratethe effectiveness of the proposed FPGA-based controlsystem. First, the experimental results of the sliding-modecontrol system are depicted in Figs. 5 and 6. The trackingresponses due to a periodic step command at the nominalcondition and parameter variation condition are depicted in

0 m position

command

mover

position

0.1 m

1 s

a

1 s

c

0 m position

command

mover

position

0.1 m

b

0 A

2 A 1 s

d

0 A

2 A 1 s

Fig. 5 Experimental results for the sliding-mode control systemfor a periodic step commanda Mover position at the nominal condition

b Control effort at the nominal condition

c Mover position at the parameter variation condition

d Control effort at the parameter variation condition

0 m

position

command

mover

position

0.05 m

0.05 m

2 s

a

0.05 m

0.05 m

2 s

c

0 m

position

command

mover

position

2 s

b

0 A

2 A

d

0 A

2 A 2 s

Fig. 6 Experimental results for the sliding-mode control systemfor a periodic sinusoidal commanda Mover position at the nominal condition


c Mover position at the parameter variation condition

d Control effort at the parameter variation condition



10/12

Figs. 5a and 5c; the associated control efforts are depicted inFigs. 5b and 5d. Moreover, the tracking responses due to aperiodic sinusoidal command at the nominal condition andparameter variation condition are depicted in Figs. 6a and6c; the associated control efforts are depicted in Figs. 6band 6d. From the experimental results, the degraded

tracking responses shown in Figs. 5c and 6c are inducedby the inappropriate selection of the lumped uncertaintybound. Although a large bound value of the lumpeduncertainty can be selected to confront the uncertaintiesthat exist in practical applications; it will result in a largechattering phenomena in the control effort. Now, the fuzzysliding-mode control system is implemented to control themover position of the LIM drive. The experimental resultsof the tracking responses, control efforts and estimation ofthe upper bound of the lumped uncertainty due to periodicstep and sinusoidal commands at the nominal conditionand parameter variation condition are depicted in Figs. 7and 8. Compared with the experimental results of thesliding-mode control system shown in Figs. 5c and 6c, the

degraded tracking responses are improved as shown inFigs. 7dand 8dusing the fuzzy sliding-mode control schemeat the parameter variation condition. Moreover, the

nonzero values of the estimated upper bound of the lumpeduncertainty at the nominal condition shown in Fig. 7c areinduced by the friction force and unmodelled uncertainty inpractical applications. From the experimental results, therobust control performance of the proposed FPGA-basedfuzzy sliding-mode control system under the occurrence of

parameter variations at different trajectories are obviousowing to the on-line adjustment of the upper bound of thelumped uncertainty.

Since the three-phase coil of the adopted LIM is veryprone to over-heating, from 251C at the beginning of theexperiment to 771C after 20 min of operation, the primaryand secondary resistances inevitably change during theexperiments. To test the control performance of theproposed FPGA-based fuzzy sliding-mode controller dueto the changes in the resistances due to heating at a three-phase coil temperature at 771C, the experimental results ofthe tracking responses, control efforts and estimation of theupper bound of the lumped uncertainty due to periodicsinusoidal command at the nominal condition and para-

meter variation condition are depicted in Fig. 9. From theexperimental results, the control efforts shown in Figs. 9band 9e are larger than the control efforts shown in Figs. 8b

0 m position

command

mover

position

0.1 m

1 s

a

0 m position

command

mover

position

0.1 m

1 s

d

1 s

c

2 V

0 V

1 s

f

2 V

0 V

1 s

b

0 A

2 A 1 s

e

0 A

2 A

Fig. 7 Experimental results for the fuzzy sliding-mode control system for a periodic step commanda Mover position at the nominal condition


c Estimated upper bound of the lumped uncertainty at the nominal condition

d Mover position at the parameter variation conditione Control effort at the parameter variation condition

f Estimated upper bound of the lumped uncertainty at the parameter variation condition



11/12

and 8e, which are the experimental results at a three-phasecoil temperature at 251C, due to the changes of the primaryand secondary resistances. However, the tracking responsesare still robust considering the change of electricalparameters.

6 Conclusions

This study has successfully demonstrated the design andimplementation of a FPGA-based fuzzy sliding-modecontrol system for the position control of the mover of aLIM drive system. First, the dynamic model of an indirectfield-oriented LIM drive was introduced. Then, a sliding-mode control technique was designed. However, the upperbound of the lumped uncertainty is necessary in the designof the sliding-mode controller. To relax the requirement forthe upper bound of the lumped uncertainty, a fuzzy sliding-

mode controller with a fuzzy inference mechanism to adaptthe lumped uncertainty in real-time was proposed. Theproposed FPGA-based fuzzy sliding-mode control system is

robust for parameter variations, external force disturbancesand frictional forces at different trajectories. Finally, theeffectiveness of the proposed low-cost high-performanceFPGA-based LIM drive has been confirmed by experi-mental results.

The major contributions of this study are: (i) thesuccessful derivation of the fuzzy sliding-mode control lawfor the LIM drive system; (ii) the successful implementationof the indirect field-oriented mechanism and fuzzy sliding-mode controller in an FPGA chip; and (iii) the successfulapplication of the FPGA-based controller to control themover position of an LIM with robust control perfor-mance.

7 Acknowledgment

The authors would like to acknowledge the financialsupport of the National Science Council of Taiwan, ROCunder grant NSC 93-2213-E-259-025.

0 mposition

command

mover

position

0.05 m

0.05 m

2 s

a

0.05 m

0.05 m

0 mposition

command

mover

position

2 s

d

2 s

c

2 V

0 V

2 s

f

2 V

0 V

2 s

b

0 A

2 A 2 s

e

0 A

2 A

Fig. 8 Experimental results for the fuzzy sliding-mode control system for a periodic sinusoidal commanda Mover position at the nominal condition


c Estimated upper bound of the lumped uncertainty at the nominal condition

d Mover position at the parameter variation conditione Control effort at the parameter variation condition




12/12

8 References

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synchronous servo motor drive, IEE Proc., Control Theory Appl.,1998, 145, (1), pp. 6372

0 mposition

command

mover

position

0.05 m

0.05 m

2 s

a

0.05 m

0.05 m

0 mposition

command

mover

position

2 s

d

2 s

c

2 V

0 V

2 s

f

2 V

0 V

2 s

b

0 A

2 A 2 s

e

0 A

2 A

Fig. 9 Experimental results for the fuzzy sliding-mode control system for a periodic sinusoidal command at a higher three-phase coiltemperaturea Mover position at the nominal condition


c Estimated upper bound of the lumped uncertainty at the nominal conditiond Mover position at the parameter variation condition

e Control effort at the parameter variation condition



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