Werner Leonhard Control of Electrical Drives

:\i\O

15

. Some Applications

of

Controlled Electrical Drives

rrcquency-

and

time- based sensing

may

be carried

out

in

paralIel, selecting

w ilichever

result

offers a

better

resolution.

The

resolution

ofthe

speed sensing

can

also

be

improved

without

analogue

i Ill.erpolation by forming in each interval

the

integer difference of

the

incre-

Illcut.al counts

and

simultaneously measuring

with

a high frequency reference

til('

differential

time

between

the

first

and

last increments [K10].

ln

the

past,

when digital signal processing

equipment

was bulky

and

ex-

jl('llsive,

it

was only used when

it

was absolutely necessary

to

achieve

the

J'1

~

qllired

accuracy, i.e. in

the

outer

position

controlloop

of a servo drive,

but

wit.h

todays

microelectronic

hardware

this is changing rapidly. AlI

the

control

rlluctions in Fig. 15.9, including

the

control

of

an

induction

motor

and

the

!('rcrence model,

can

now be

incorporated

as a

modular

algorithm

in a micro-

cOlllputer which finds room on a

postcard

size

printed

cardo

If

necessary,

the

s;

ullpling

time

in

the

order of 1 ms

can

be reduced

by

a factor of

ten

when

(,Illploying signal processors processors; hence DC servo

motors

are

rapidly

sllpplanted by digitally controlled

and

compact AC drives.

This

is a

remark-

;d)!c development

made

possible

by

the

joint

advances of microelectronics

and

jlowcr electronics.

It

was mentioned before

that

backlash in gears

and

couplings

may

cause

jlroblems with position controlled drives because

the

contact

between two

;t.djoining

parts

of

the

drive is

temporarily

lost when

the

transmitted

force

is rcversed; as a consequence

the

surfaces formerly exerting opposite forces

;\II~

llloving

apart,

until

contact

is again

made

at

the

other

side of

the

gear

l.ooLIt.

Clearly

this

discontinuity

of

the

transmitted

force, possibly combined

wi

I.h

Coulomb friction, is in conflict with a

smooth

and

precise position con-

I.ml

[1365].

Backlash

can

be

avoided by employing high quality mechanical

("()I

II

jlOllents,

but

wear

cannot

be

excluded. A mechanical solution would

be

1111'

IIse of spring-Ioaded gears with split wheels which

maintain

a

bilateral

(·olll.ad force irrespective of

the

sign of

the

transmitted

load force.

AIlother solution requiring only

standard

gears is

to

split

the

drive into

:

;

('V(~ral

Illlits

and

to

preload

the

gears electrically;

this

is

of

particular

interest

ii'

til('

type of load makes a

distribution

of

the

driving force desirable anyway.

Ali

(~xample

could be

the

azimuth

motion

of a large satellite

antenna,

i.e.

the

j'()l.al.ioll

around

the

vertical axis;

antennae

of

this

type

require very

accurate

jlosil.iouing, for example with a final tolerance of less

than

1/100°.

This

rorn:spollds

to

a resolution

of

the

fulI circle

to

one

part

in 36,000 which

ddillitely

calls for direct

measurement

by a digital sensor

mounted

at

the

pivoL

!>caring

of

the

main

construction.

The

two or

more

servo motors could

I

II

:

arranged

at

symmetrical

points

arollnd

the

circumference

and

exert torqlle

I.hmlll·;1t

pillioll drives to

the

central gear. A simplified schematic is drawlI ill

I"i/':. I r

i.

1:\

ass1lmiu!; two motors,

with

the

main

struetur

e of

thc

étutellUi1

(tlld

1.11('

I.wo

drivI~

lIIoLors

I>eillg

IUlI1ped

illto

thr<'(~

sqmrate

rigid

I>o<lie:;

. TIl(:

1'l1.rld;ll

:h

1I:

,:

';

IIIII('d

ill

j.

)()l.ill~('nr

I.raills is lIIadc illdr('I,t,iv(:

by

<I,ddi1lg

opposit,,!

"il

H

ei,

l

ii

l

l',II

'

''

':

Lo

1.11(1

I,(lrqll('

n:fI:I'(:)l(

'Ps; I,!tis

1'('::1111.1'1

ill

jll'('loadiul':

1.11("/('

111

'11

iII

15.2 Linear Position

Control

381

the

low

torque

region, where backlash would

be

most

troublesome. As

the

required load

torque

mM

Ref

increases,

the

motor

that

initially opposes

the

motion, eventually reverses its

torque

and

comes

to

the

help

of

the

other

motor

until

there

is nearly equal load

sharing

at

full torque.

Centralload mass

m

c

t,

ro

l,fl

~

m

C

2'

ro2'

f2

Motor 1

Motor 2

Fig.

15.13.

Position drive

with

two servo

motors

and

electrically

preloaded

gears

for

counteracting

the

effects

of

backlash

When

assuming

symmetry

between

the

two motors, neglecting ali fric-

tional

torques

and

referring all angles,

angular

velocities,

inertias

and

torques

to

the

central

axis,

the

block

diagram

in

the

right

hand

side of Fig. 15.14 de-

scribes

the

mechanics of

the

drive.

The

torque

control loops

are

again replaced by equivalent lags having

unity

gain.

ln

addition

to

inherent

damping

effects

not

shown in Fig. 15.14,

such as speed

dependent

frictional torques, active

damping

will definitely

be

needed

with

this

complicated mechanical

structure.

Two

kinds

of

oscillations

are

likely

to

require countermeasures

Gears wi/h elasticity

and

backlash

Fig.

Hí.'14.

Silllplificd block diagram of three-mass

twin

drive

with

preloaded gears

for

CO\lllt(·l'Ilc'l.ill/',

hacldlLHh

:\

X:2

15. Some Applications

of

ControJled Electrical Drives

• oscillations of

the

two drive motors in phase opposition while

the

central

mass

remains

at

rest,

• oscillations of

the

two

motors

in phase against

the

central

mass.

Provided

the

torque

control loops respond sufficiently fast,

both

oscilla-

I.ioll

s

can

be

damped

by adding suitable signals

to

the

torque

references,

LlmMl

Ref

= -

DI

(Wl

-

W2)

- D

2

(Wl

+

W2

-

2w)

,

LlmM2ReC=

D

I

(WI-W2)-D

2

(Wl+W2-

2w

).

T!te signals for

the

motor

speeds

w},

W2

can be derived from sensors

attached

1."

the

motor

shafts.

It

is

noted

that

the

steady

state

characteristics of

the

dri

ve

system

are

not

altered by these auxiliary feedback loops because

the

~il-!;Ilals

vanish

in

steady

state

under

symmetrical conditions [T31].

Fig. 15.15 demostrates

another

pair

of nonlinear functions

that

would be

a

ti

va

ntageous for

the

preloading of gears because they cause the

motors

to

produce equal torque

at

fullload

while maintaining

constant

overall gain in

I.!t(! speed control loop. Again these nonlinear functions would be cumber-

~O

lllC

to

realise with analogue circuitry

but

they

present no problems when

Illicrocomputers are employed for

the

controI.

/

/----

/-mM

tRel

+

mM

2

ReI

mM

ReI

/

"' il~.

tó.15.

Torque

functions for electrically preloading

of of

gear

train

t\llother example of a dual axis position controlled drive is shown in Fig.

I;).

1fi a.

It

could be

part

of

a machine tool or a

robot

with a mechanical

:

,t

rllcLul'c in cylindrical coordinates r,

f,

z, where each axis

is

activated by a

:;

('par;\I

:e

position controlled drive having

an

inner

torque

controlloop

with

!.II(!

eqllivalent lag Te. This

is

a simplification of

the

example discussed in

(:hap.

2.

For

th

e

particular

application

it

could be necessary to generate

I.!t,

!

r<'f(,fCll

ce

trajectory

in

x,

y, z-coordinates, while only machine

orientated

'I','

,

;;

-

1I1

Ci\S

Ur

ements are available. Clearly the motion in

the

z-direction is de-

,

'0

11

pkd

ir

it

is vertically

orientated

so

that

gravitational

effects are exclndcd I

1>11

t

ii,

trallsfornmtion bctwcen

(7'

, f) and (x,

y)

is

nceded.

It

can be proclll'ecl

I,,Y

I.

r

<i.

wJOrl

IlÍlI

I,

':

t.he

:G,

y··re

fc.

rence data.

int:o

polar

coordinat(~:;

or

iuv(

:rsely by

('()lIv"rl

,lrq':

t.\1('

'

I'

, ,,·

f(

!,!

dhack siguais iuto

carl.(!~jall

('()()rdilliLt

cR

j

I.h

e lirR

I.

P08-

lI

llti

lll.y

itl

('

h"

H"lI

for

1,\11'

1Ilino

l'O

lllplll,"I'

hIl

H,'

,j

(,oll!.rol

:i

l'l!"Jlll'

iII

FiJ~

,

1r" W

I,

h\

'

(

'

!l ll

tI

~

'

a

li

I.Iw illlll'r

"<lII!.r,,1

1I

IIII,·l.i'lIlI

:

an

'

1.111'11

I)(T

J'

"rl)lI'd

1

11

1.111'

a('

l.llnl

15.3 Linear

Position

Control with Moving Reference

Point

383

Y~

M

'

~

1

mM

E

a

x

Contjnuou~

I

Coord

inate :

Motion in

, Motion in

r

Bfe

renc~s

m: transfo

rmat

ion I

Decoupled

conlrol

cartesian

cartBSian

I :

polar

coordinates

f

in

polar

coordinatBs

: coordinatBS

coord

inates :

~

!

,

~

b r

ro2

8C:C

~

::~~ff~

,

I

:

'

l;;+;t--~,~r

,

·~t

--~~""'I

I

JRof

I +

,

r

lx<+i'

-

,

!t:ir

-ll--=-r,~"---

I

x<+i'

arctan{

b

Fig.

15.16.

Dual

axis

position

control in

polar

coordinates

with

reference signals

supplied

in

cartesian

coordinates

(

a)

Mechanical modelj

(b)

Block

diagram

drive coordinates.

The

set (x, y, Z

)Rej

would again be produced by a

suitable

reference

generator

.

Dynamic

coupling effects between

the

r,

f motions caused by centrifugaI

and

Coriolis-forces

can

be approximately compensated by

adding

suitable

corrective

terms

to

the

torque references, as was mentioned beforej

the

ef-

fectiveness of

this

method

of compensation, called inverse load modelling,

depends of course on a fast response of

the

inner

controlloops.

With

the

present

state

of microelectronics alI

the

control functions seen

in

Fig. 15.16 b

can

easily be executed with 1 kHz sampling frequency, which

would be

adequate

in

most

cases.

15.3

Linear

Position

ControI

with

Moving

Reference

Point

ln

Sect. 15,1

it

was mentioned

that

there

are cases, for example with

paper

mills, where

the

speed of each drive section

must

be controlled

with

high

pr

ecision in relative synchronism

to

the

speed of a neighbouring section.

This

task cnu

<tIRO

I)c ronsidered as a position control problem, where

the

reference

po!';iLi()1l

chall

lçl's

iII

sLeal\y st.at(· nt

CO[lst

n.

nt

vd

ocity,

As

an

exa

mple

the

di,c

il.

nl s

p(

·(·

d

nJIILl'ol

~

w

li

f

~JlI

(,

in

F'i,(,

1 :

1.1

'1

wl,i

l'

h Im

ll

h

(,

,

~

lI

d('vd()ppd prior

Lo

til<'

II

dv"liI, "r

IIl

kr

'

I\'I

I'

('

I,

rlll

dni

JlI

II

.y )1(' b

rk

lly d ili

r\l

H/

'h

'd

,

384

15.

Some

Applications

of

Controlled Electrical Drives

lt

is based on a conventional analogue speed control loop using, for ex-

ample, a

De

tachometer as a speed sensor. Due

to

unavoidable drift of

the

analogue components there is a normal control error, usually below

10-

2

of

Ilominal speed, which could, of course, be reduced by one or two orders of

Illagnitude,

but

only with considerable effort, such as

thermal

stabilisation of

the criticaI components,

and

at

high cost.

The

idea

of

the

scheme in Fig. 15.17

is

to

add

a digital corrective channel

for compensating

the

steady

state

error of

the

analogue speed control loop;

rapid

response

is

not

important

since a well-tuned analogue speed control

leaves little

to

be desired in

this

respect.

ZI

Inkremental

sensor

Digital

controller

Fi

g .

15.17.

Quasi-continuous digital speed control

The

digital controller compares

the

pulse

train

h from

an

incremental

S

('IISor

, where each pulse corresponds

to

an

angular increment

.dE,

with

a

Il'krence pulse

train

fI

derived from a high accuracy frequency source. A

I(

~

v(

~

rsible

counter registers

the

forward

and

reverse pulses; if none

is

lost,

the

t'()I\Lcnt

of

the

counter is a measure of

the

angular error

of

the

drive shaft

wiLh respect to the reference angle.

With

the

help of a digital-to-analogue

COIl

verter

it

is

transformed into a small corrective signal

.dWl

for

the

analogue

Sl)(~ed

loop. Hence

the

digital channel may be

understood

as

an

integral speed

conLroller

without

drift or as a proportional position controller [J24,

L31]

.

15.3

Linear Position Control with Moving Reference Point

385

Zl

Overflow

z(v+1)

Fig.

15.18.

Programmable frequency divider

ln

order

to

generate a finely adjustable reference frequency h having

the

sarne

steady

state

accuracy as

the

fixed clock frequency

io

derived from a

Quartz

oscillator,

the

programmable

frequency divider shown in Fig. 15.18

might be used.

It

contains

an

arithmetic

unit

for executing a recursive addi-

tion

according

to

z(v +

1)

= [z(v) + zd

mod

Zmax,

(15.6)

where

Zmax

is

the

capacity of

the

register while Zl <

zmax

corresponds

to

the

desired reference frequency

fI

established by

the

overflow pulses.

The

addition

is

synchronised

with

the

clock frequency

io.

The

mean

balance of

the

continued

addition

is

fI

zmax

=

io

Zl

so

that

a pulse

train

results,

the

mean

frequency of which is

adjustable

in

fine steps

and

has

the

sarne precision as

the

constant

clock frequency

io,

Zl

fI=

-io·

(15.7)

Z

ma

x

The

overflow pulses

are

not

quite equidistant

but

when

operating

the

fre-

quency divider

at

a sufficiently high clock frequency,

this

inherent phase

jitter

has no effect on

the

speed of

the

motor

[L32].

Similar

methods

are used in

the

process

industry

for in-line blending,

where

the

flow

rates of liquids

to

be mixed in prescribed ratios

are

measured

by

tnrbine

wheel

flow

meters whose speed is

transmitted

and

processed in

the

fmm

of

plll

s

(~

trains

[R12].

A1\ adjllsl.;\,hi<-

n'fcr<~llCC

frc(jll(:llcy

lJlay

abn

llC

formed with the

help

of

;~

plta

:w I<lc\\(,"

1<I<lp

(PI

,I

,)

1t;l.ViJI/

( H

~

i('

;d(

'r

wil.h

t'a.pa.<:ity

'::

1

iu

r.h(~

k c

<!hack

paLlI;

LI

,j

fi

pr"

"

111

,, ';1

aI.

LI\I'

,,"LplIl.

"I'

II

v"I

I.I

I

-/,

," n>lII.I'<lll,'d w:,·

ill

ll.

l.<l1'

(Vi

~O)

:1

86

15. Some Applications

of

Controlled Electrical Drives

ii

multiple of

the

constant

clock frequency, f =

zlfo,

which

may

then

be

rcduced by a scaler with capacity

Zmax'

ln

various production lines where continuous strips or webs

ofmaterial

are

hcing processed in subsequent

stations

it

is

often specified

that

the

stations

.

~hould

be decoupled for preventing tension

in

the

material. For this purpose

a limited

storage

of

material

is

accumulated in

the

form of a loop between

t.he

stations

which, by

taking

up

the

difference of incoming

and

outgoing

IIIaterial, represents a measure of

the

synchronism between

the

two adjoining

drives. Naturally, one should

attempt

to

maintain

the

loop

at

approximately

co

nstant

mean

length to have adequate reserves in

both

directions, should

t.h

e speeds

of

the

two drives exhibit

temporary

fluctuations.

This

is

illustrated

in Fig. 15.19 a showing

the

principIe of a magnetic

tape

drive as used on computers.

The

incoming

tape

is fed from a reeI

and

pressed

a

.g

ainst

the

drive rolls

AI,

A~

to

be

transported

with

high acceleration

and

velocity

past

the

read-write

head

S;

subsequently

the

tape

is rewound on

a

nother

reeI driven by a

separate

motor. By keeping

the

tape

between

the

reels

taut,

the

maximum

acceleration would be severely limited by

the

inertia

of the reelsj also, excessive force

in

the

tape

could

not

be

excluded.

ln

order

1.0

avoid these problems, loops

are

formed

on

both

sides of

the

drive rolls

which are kept staight with

moderate

forces by mechanical or

pneumatic

equipmentj

this

greatly reduces

the

effective

inertia

and

the

applied forces,

when

the

tape

is

accelerated.

The

more slowly accelerating reeI drives

must

of course produce sufficient

maximum

speed for preventing

the

storage loops

from reaching

their

upper

or lower limito

a

VI

Feed - forward

\

.

I

.----..,

I

-I~I

1

./

ali

l:..i

m

L

:

a2

X2

v

2

----

mI\1R~A:11

/)

Jo'ir

~

.

1r;. I{lo I'

·h

:d d ri

v"

for

maglldi

c,

t.

npc

(1\)

Schcllml

,i

cj

(J.)

Blnel

, diugrn.m

15.3 Linear

Position

Control

with

Moving Reference

Point

387

Only

the

left

half

of

the

tape

drive

in

Fig

. 15.19 a

is

considered

in

the

following, with

the

tape

velocity

VI

at

the

read-write

head

S assumed

to

be

an

independent (impressed) variable.

The

speed

V2

of

the

feed rolls should

be

so controlled as

to

maintain

the

length

X2

of

the

storage loop

at

a prescribed

vaI ue. Because of

dX2

= t (V2 -

vd

(15.8)

dt

the

block

diagram

in Fig. 15.19 b results, where

the

storage represents

an

integrator

with

two

inputs

cancelling in

steady

state.

Normally

the

feeding

reeI

is

controlled

to

maintain

constant

tension of

the

unwinding

tape

.

The

quality

of

the

control can be further improved by introducing

to

the

speed controller of

the

feed rolls A

2

a feed-forward signal from

the

velocity

VI,

which acts as a disturbance.

ln

view of

the

quickly changing

operating

conditions

it

could even

be

appropriate

to

provide acceleration feed-forward

to

reduce

the

necessary length of

the

storage loop.

It

is noted

that

the

in-

ertia

of

the

reeI drive depends

on

whether

the

spool is full or nearly emptyj

also,

most

magnetic

tape

drives are reversible, i.e. they can

operate

in

either

direction.

I

Ifm

b .1. b

Fig.

15.20.

Geometry

of

loop

in

a continuous rolling mil!

On

multi-stand

continuous rolling mills

the

storage loops between suc-

cessive stands, which are necessary

to

trim

the

coordinated speed control

described in Sect. 15.1, have a different geometry, as

is

shown in Fig. 15.20;

an

electrically or hydraulically powered looper roll pushes

the

material

with

a force

h upwards

and

thereby keeps

the

strip

taut

. Due

to

the

geometry

of

th

e loop

there

is

a nonlinearity between

the

deflection x

and

the

length

of

the

loop, I = .Jb

2

+x

2

leading

to

a nonlinear differential equation

2b::.d(xjb)

V2_VI

(15.9)

Vo

I

dt

Vo Vo

whc

r<:

VII

coulel oe

th

c

rated

velocity;

the

integrating

time

constant

'J.

ii :1:

T,

(I~U())

'

/I"

I

389

:\

88

15. Some Applications or Co

ntroll

ed Electrical

Dri

ves

dcpends on

the

ratio

x/I, i.e.

the

shape

of

the loop.

x/L

= 1 corresponds

to

the

special case in Fig. 15.19 a.

Operating

the loop

at

smaU values

of

x / l becomes

increasingly difficult because

the

integrating

time

constant

is reduced;

this

calls for fast response

of

the

looper

and

the adjoining main mill drive used

for

controlling

the

loop.

The

force

1m

exerted on

the

material

is aIso a function

of

the loop geom-

ctry; in

relation

to

the

vertical force h exerted by

the

looper roll,

it

is

1m

1 I

(15.11)

IL

= 2 x '

;dso indicating a criticaI range

at

small values

of

x/I, where

the

gain becomes

very large.

A similar problem as with loop control arises with

automated

tracked

vchicles developed in recent years for

urban

transportation,

Fig.

15

.

21

a.

The

dis

tance

x

of

each vehicle to

the

leading vehicle

must

be

controUed

to

a speed-

dependent safe value

XRef

by applying driving or braking force

1M

~

O.

The

distance can be measured, for example, by

radar

or by sensing equipment

ill

stalled in

the

guideway.

--~-

-- - - - -

------,,--

f

M

'L

X

a

f

L

VI

b

Fil-:.

15.21.

Distance control

of

tracked

vehicles

(1\)

Sct.cma

ti

c;

(b)

Block

diagram

WIWII again

n

~

g

arding

the vclocity of the forward vchicle as an indepen-

dl:lI1.

di

s

l.llrl,all(:(

:, tlw control systcm for the distallce between the vc

hidcs

("(

)illd h

av

I:

t.l1('

fOl"I1\

ShOWll

ia

Fi

g. 15.

21

b. li:

J"('pr

c:;e

n(.

s a cascadcd

s

dl(

~

Jll.t:

"urIÜ t

illilll'

:

il

\l

l<'L"

1<l

l/

ps ror

UI(

: dri.v

i.

ug

r()r

c

~:

a

lI"

vdo

city.

Th(~

d('.Hirc,1

"i

s

t

ll

ll'·.·· .

1'1

1,.,

1,0

U

H'

"'/".d iu

l'.

v

dliclc

in

1'

:

('n

•.

'

l'I

l.

L ~

'

d

mi

l\

d,·lny,'d III

Ild

i

11(';1.1

.'

J'lIllcLÍ.OII

" r V<'!f1('1

I.

V,

1

11

1 "

KI

II

II.

I"I

,· II

I'

ll r,ll! H

,J

'1 1

\.j

,l

l'

o

vi

lJl

lI

J.l

III\ Il l

li1

r

..

" r!

t.l

d

ll

l'.

di

:il.ll.lIl ·l '

15.4

Time-optimal

Position Control

with

Fixed

Reference

Point

v

km/h

120

60

..

_--_

......

....

041

------,-

--

----,-------

__

o

10

20

tis

x =x =

SOm

Rei

'

........

\>,

"'-"""'"

f

MRef

"

O-t

O

--

...

-"

20

tis

Fig.

15.22.

Braking

transients

of

distance control scheme

A

computed

transient

of

the

distance control scheme is shown in Fig.

15.22;

it

may

be

caused by a sudden constant deceleration

of

the

leading

vehicle; a PID-controller

and

a linear function

XRef(V2)

were assumed for

the

example. Transients of this sort

and

their

dangers are well known to

the

motorist;

the

task

of

maintaining

a safe distance can be alleviated by

employing advance signals, such as feed-forward from

the

deceleration

of

the

leading vehicle

or

the

rate

of

distan

ce change, as is common

pra

ctice in

road

traffic by use

of

braking lights

or

warning signs.

15.4

Time··optimal

Position

ControI

with

Fixed

Reference

Point

Some applications

of

drives call for

the

discontinuous motion of a machine

part

from one

steady

state

position to another. Examples are boring machines

where

the

tool

must

be

positioned for

the

next drilling operation

or

reversing

rolling mills where

the

upper

rolI is lowered by a screw-down drive as soon as

the slab

of

ma

terial

has left

the

roU

gap. Since

the

load

is

disengaged

during

th

e

tr

ave

I,

thcr

e is no specified

traj

e

ctory

x(t) ;

the

only requirement is

th

at

tIl<:

driv

(~

sllOlIld follow tlw disc

ontiJlllOIlS

position reference in minimum

tim

e

l:

iv

('" I.he

po

wer

ra

.t.illl';

nr

til(' driv(' alld w

il

.hollL ovr rs

hooting

th

e

targ

c

l:

pO:

ií

l.illll

,

:\90

15. Some Applications

of

Controlled Electrical Drives

With

the

assumption

that

•

the

response

time

of

the

torque control loop

can

be

neglected,

•

the

jerk, i.e.

the

derivative

of

acceleration, need

not

be

restricted (which

would

be

necessary if a

transportation

system

with

on-board

passengers

or

a weakly

damped

mechanical

structure

were involved),

•

the

frictional torques are ignored,

I.he

drive

with

the

associated mechanicallinkages can

be

approximated

by a

double

integrator

having

the

acceleration a(t) as

input

and

the

position x(t)

itS

output,

as seen in Fig. 15.9.

The

well known time-optimal solution

that

could

be

derived

with

the

cal-

culus

of

variations, consists

of

two intervals

of

maximum

accelerationj

deceler-

ittion

and

possibly an intermediate interval with

constant

maximum

velo city.

This is shown

in

Fig. 15.23 for two different step changes

of

the

reference po-

sition [38,44]. Velo city

and

position are

plotted

versus

time

as well as

in

state

space which, because x

and

vare

the

only

state

variables, reduces

to

a

state

pl

ane. Beginning

at

t = O in

steady

state

condition,

v(O)

=

x(O)

=

O,

a new

position reference is commanded,

XRef

=

XOl,

calling for a new

target

state

'II

(t

)

=

O,

x (te) = XOl

with

te

--+

mino

The

time

optimal

transient consists

of

e

lII

ax

imum

initial acceleration,

operation

at

maximum

speed

and

controlled

deceleration

to

the

prescribed

target

point;

the

maximum

values

of

acceler-

ation

and

deceleration are assumed equal in Fig. 15.23.

If

the

amplitude

of

t,h

e position step is reduced,

X02

<

XOl,

the

maximum

velocity may

not

be

r('ached;

the

acceleration is

then

immediately followed by deceleration with

I.h

e switch- over taking place

at

half

the

distance travelled.

v-

Ti<

a

ma

x

-a

max

t

3

v

X0

1

x

X

OI

FI

l{.

1r .

...

~

:L

Id'·ldin,

·,1

Lim

.,'-op(I,iJ

lIl..!

1"'

~II

,

ionill,

{

!.n l

ll

!l

;"

U

(,

wil.h lilll i

kd

1l

c<~

c:I"

r1~

l.i

o

ll

(n)

~

H,

J

t..

h

·

1'

1t

111t

'1

(J.,

)

1t1'

)l t

)

(

IH

I

U

~

vr

f.

1,llIl

n

15.4

Time-optimal

Position

Control

with

Fixed

Reference

Point

391

f

L

Fig.

15.24.

Time

-

optimal

position control scheme

ln

the

state

plane,

the

intervals

of

constant

acceleration

or

deceleration

are

characterised by

parabolae

marked

with

time,

the

interval

at

constant

velocity

appears

as a

straight

line parallel

to

the

x-axis.

The

multi-Ioop position control scheme

in

Fig. 15.24 closely models

the

idealised behaviour shown in Fig.

15

.

23

[A24]

.

The

nonlinear position con-

troller

may

be

realised digitally

to

achieve reproducible accuracy

at

high

resolution, for example 0.1

mm

with a

maximum

distance

of

1 m on a screw-

down positioning drive.

The

scheme differs from

the

linear one

in

Fig. 15.9

by

the

nonlinear function in

the

position controller,

VSoll = v 2 a

m

ax

lei

sign

e,

(15.12)

which generates

the

velo city reference as a function

of

the

remaining position

error

e;

its

purpose

is

to

guide

the

drive

with

constant

maximum

decelera-

tion

into

the

reference point.

The

mode

of

operation, after a

step

change

of

the

position reference has occurred, is

then

as follows:

When

XRef

changes

by

avalue

exceeding twice

the

maximum

braking

distance,

the

velo city ref-

erence

VR

ef initially assumes

its

maximum value V

max

causing

the

velocity

controller

to

become

saturated

and

the

drive

to

start

at

maximum

acceler-

ation.

When

the

velo city reaches

the

reference value,

the

velocity controller

reverts

to

linear o

peration

and

maintains

maximum

velo city. As

the

position

error

is

gradually

reduced, VRef(e) follows

the

parabolic function, resulting

in

constant

deceleration until

the

reference position is reached.

The

drive

may

then

be

disconnected by disabling

the

power supply

and

arrested

with

a

mechanical brake. This is necessary because

of

instability due

to

the

infinite

gain

of

the

position controller

at

e

~

O.

ln

case

the

position reference changes

by less

than

twice

the

maximum

braking distance,

the

drive will switch from

acceleration

to

deceleration before

the

maximum

velo city is reached.

The

torque

control loop in

Fig

. 15.24 serves only for protection in case

of

overload;

the

rnaximum current should be

adequate

to

produce

maximum

ace

e!

craLion at

fnU

load torqlle.

1\

lIo(.\wr

1I1

()

f(

~

<'!

e

f.Ç

i.t

Ilt

llldh

od

for

n'

a.li

sing lninirnum time position con-

l.rol ili ·,d" 'w

II

i

II

II'i

,

~

.

I

fi,

2;' ,

wh(

~

J'(~

i.l\(

~

lill<'ilí'

rd crf'

ll('(

' Kiglln.l

w

~ncrnL()r

of

I,'il',' I

~,

.

H

1,

1111

1"

,(,

11

1I""lili"d

l."

,',

"1\1'1'

:.1

.1.,

' 1.

1",

r,

',·

,I

-r

"rw

nr,1 ;;i,.,lI i

t.l:

l

)'I"llIi!"

'

,1

:\!)2

15.

Some

Applications

of

Controlled Electrical Drives

I

-.

X

7

I~arget

X

Flef

V

Ref

a

Ref

Fig.

15.25.

Dynamic reference

mo

dei

for

minimal-time position control

ror

time-optimal

transients.

The

advantage of

this

arrangement is

that

the

coutrol system proper remains unchanged, i.e. linear

and

optimally tuned.

'I

°lte choice, whether

the

transients

should

be

linear

or

time-optimal is solely

ddermined

by

the

reference model; this

structure

is particularly

suited

for

illlplementation

on

a microcomputer.

As a

further

variation,

the

nonlinear function

in

Fig. 15.25 is modified

around

zero (in dashed lines)

to

prevent overshoot

and

to

achieve a

stable

nmtrol

mode

at

the

reference position.

The

advantages

of

both

control

strate-

l\

ies are

then

combined: For large position error,

the

drive performs nearly

I.ime-optimally while

it

becomes linear once

the

position error

has

entered a

Ilarrow range

around

steady

state

condition;

the

sarne could be applied to

Lhe

position controller in Fig. 15.24.

It

has

been mentioned

that

the

second order approximation is only appli-

cable

as

long

as

the

residuallag

of the torque

controlloop

can

be

neglected.

'I

'hus

t.he

application of this control scheme is restricted

to

converter supplied

~

Rel

!

I t

Fttrr

I

I I

~

I

t

~

I

t

I I

-r;-

tt.

IVX

~

I

.II"

I

~Posítion

v

~

v~

x

X

Slnl

e

plano

~

F

ig

. H •. :.W. M inll.II"

.!

1'.il

lL

'·

I'0 ~

il.i,,"i"

g

1.1'Il1I

::

i'·IIt.

1I

wi

t.h

difr"lt

lld.

ill

it.il\l (·

olldil.ioIlH

15.4

Time-optimal Position Control with Fixed Reference Point

393

DC

or

AC drives having

an

effective lag

of

the

torque loop of a few

ms.

A

time-optimal

transient

of

an

AC servo drive was shown in Fig. 14.9; some

simulated transients, assuming different initial conditions, are

plotted

in Fig.

15.26.

v(x)

Speed restríction v

max

at

x > x 1

Braking

curve

------,

I

I ,

I

I ,

I "

I

I "

I

:

I '

\ \

I

e

~!-v

'\

1

",,,

~~;:x2

x=o x

x2

x

1

Position

of

traín

at

begín

of

brakíng

Fig.

15.27.

Controlled deceleration of automatic train along a veocity profile

Controlled deceleration pro files are also used on

other

drives such as ele-

vators

or

automatic

trains,

where

at

any

time

a fictitious braking curve v(x,

t)

must

be

known, along which

the

vehic1e could

be

decelerated or

brought

to

standstill

with

the

maximum

specified deceleration.

By

continuously compar-

ing

this

projected velo city profile

with

possible forward speed restrictions,

it

can

be

determined, when

braking

should begin

to

meet

the

constraint. Fig.

15.27

illustrates

the

situation,

where

an

automatic

train

with its

projected

velo city profile approaches a restricted speed zone.

Whenever passengers

on

board

vehic1es are involved,

it

is usually speci-

fied

that

the

accelerating forces

must

change gradually

to

avoid discomfort

or

anxiety (as every passenger

standing

in

a bus knows)j a similar

situation

exists, when complex mechanical

structures

such as satellite aerials

or

robots

are

to

be

moved, because

abruptly

changing acceleration

or

deceleration cre-

ates

unnecessary forces

and

oscillations. This calls for

smooth

but

swift posi-

tioning, still using all

the

available drive power. Some idealised time-optimal

positioning

transients

with limited rate-of-change

of

acceleration (jerk) are

drawn

in

Fig. 15.28, beginning

with

large change

of

position reference (a)

which is

gradually

reduced

to

case (e); (b)

and

(d) are limiting cases sepa-

ratin

g differe

nt

modes of operation. Clearly,

the

condition

that

da/dt should

be

finit.c

~

muo

unts

to

th

e definition

of

acceleration a(t) as

another

continuous

st

a

tp

va.riahk

j IWllc

e,

the s

tate

v(

~

ctor

[a(t), v(t), x(t)] could be represented

as

a

cOIILillll()II:

4

I.r;l

,

j""

.I.ory

ill

1t

1'.

111"('(

' dillH'Jlsiolla,1

Spi\.

CP

.

Werner Leonhard Control of Electrical Drives

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