Pulse-Width-Modulation Control for Second Order Plants Via a Quadratic Criterion

Theses

Thesis/Dissertation Collections

1972

Pulse-Width-Modulation Control for Second

Order Plants Via a Quadratic Criterion

Charles Gebo

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PULSE-WIDTH-MODULATION CONTROL

FOR SECOND ORDER PLANTS VIA A

QUADRATIC CRITERION

by

Charles H. Gebo

A Thesis Submitted

in

Partial Fulfillment

of the

Requirements for the Degree of

MASTER OF SCIENCE

in

Electrical Engineering

Prof.

Robert E. Lee

(Thesis Advisor)

Prof.

Harvey E. Rhody

Prof.

George A. Brown

Prof.

Walton F. Walker

(Department Head)

DEPARTMENT OF ELECTRICAL ENGINEERING

COLLEGE OF APPLIED SCIENCE

ROCHESTER INSTI'I'UTE OF TECHNOLOGY

ROCHESTER, NEW YORK

ABSTRACT

This paper presents a mathematical _study of the control of

a second order plant

having

real distinct roots

by

use of

pulse-width-modulation. An integrator connected between the

pulse-wldth-modulator and

the

plant is a fixed element of

the control

loop.

The integral-square err.or is used as the

Index of performance. The ISE is minimized with the con

straint that the pulse modulator is limited to a maximum of

one output reversal for a _step change in

input.

Derivation of the equation which will predict the exact _switching time

of the modulator output to minimize the performance index

for any step amplitude is presented. A3n example problem is

worked to illustrate the use of

the

switching equation in

determining

switching

time.

The example is concluded with graphs _showing the optimal plant response for various _step

input _amplitudes, variation of _switching time as a function

of _step amplitude, and variation of

integral-square

error as a function of _switching

time.

TABLE OF CONTENTS

Page

List of Tables

-vi

List of Figures vii

I,

Introduction

II.

General

Theory

A.

System Description

B. Derivation of Plant Response Equation

8

C. Investigation of Plant Response 12

D. Formulation of Integral-Square Error

21

E.

Derivation of the

Switching

Equation

27

TABLE OF CONTENTS

(CONT'D)

Page

A.

Objective and Plant Description

35

B.

Calculation of T

35

s

IV.

Conclusion 50

V.

References 52

VI.

Appendix

53

A.

Derivation of

C,(m,T_tt)

in Terms

j s

of

c2(Ts,t)

53

B.

Evaluation of

~.M *>

u r

f

//

TABLE

(CONT'D)

C,

Evaluation of

7r

Page

fe^WU*

67

D. Evaluation of

A^r

cj

^7?^

tftt 78

LIST OF TABLES

Page

Tables

1,

2,

3

- Tabulation _of

Switching

Equation Coefficients

29, 30,

31

Tables

kt

5,

6

- Evaluation _of

Switching

Equation Coefficients

36,

37,

38

Table

7

- Optimum

Switching

Times ,

^0

LIST OF FIGURES

Page

Figure 1 - Block Diagram _of System

7

Figure 2 - Plant Response 16

Figure

3

- Optimum Plant Input 18

Figure

k

- Flow Chart for Solution _of

Switching

Equation 3k

Figure

5

- _Optimum

Switching

Time

kl

Figure

6

- Comparison _of Optimal and Sub-Optimal Plant

Responses ,

k3

Figure

7

- Optimum Response Curves

,

kk

Figure

8

- Variation _of Integral-Square Error _with

LIST OF FIGURES

(CONT'D)

Page

Figure

9

-

P.

W.

Modulator Output & Plant Input for

Input

Step

-

0.5

^8

Figure 10- _{P. W. Modulator Output & Plant Input for}

the same size motor can be used to operate several different sized valves. This reduces

the

spare parts

inventory.

A typical motor drive device for use with valves consists

basically

of an AC motor, gear

train,

limit switches and a

slidewire which is driven from the output shaft. The motor

used in this drive will reach full speed from rest or go

from full speed to rest in the time required for one cycle

of the applied voltage

(i.e.

1 of a sec.). This response is _extremely fast

by

comparison with the speed of response

of most processes and for all practical purposes the motor dynamics _may be neglected. The nonlinear characteristic

of the valve can be compensated

(i.e.

linearized) by

adjust ment of a crank and linkage mechanism on the drive unit.

Therefore,

the combination of the drive unit and valve can be regarded as an

integrator.

The standard approach to _controlling systems

having

electri

cally actuated drive mechanisms evolves around the classic three response controller with a slight modification, A dead band circuit is added between the controller output and the motor _actuating relays and the controller feedback signal is taken off the slidewire wiper instead of from the controller output. The dead band circuit is used to prevent small random changes in plant output from _continually turn

The basic

difference

between an electric valve drive and a

pneumatic diaphragm motor drive is that the latter is a con-.

tinuous

device

whereas the former is an on-off type

device.

Taking

the controller feedback signal from the slidewire

wiper in an electric motor drive system is an attempt to

eliminate the on-off nature of the drive and make its res

ponse more like the pneumatic

device.

This goal is achieved

to the extent that it is possible to control the average rate

of valve

travel.

Assume that

the

controller integral response

is adjusted so that the rate of change of voltage at the

slide-wire wiper generates a feedback signal large enough to cancel

the difference between the reference input and plant output.

Under this condition the motor will _stop and remain at rest

until the decrease in feedback voltage allows the controller

output to exceed the dead band limit and turn the motor on

again. The motor on-off sequence is continued until plant

output and reference signal are in agreement. The valve move

ment is a quasl-staircase function and it can be seen that the

average valve velocity can be controlled

by

_adjusting the

controller integration rate.

Loops are _generally tuned _according to the 1/k

decay-ratio

criterion. This means that the process

loop,

and

therefore

the motor and valve, will cycle until the plant reaches

the

wear on

the

motor, valve, and other mechanical parts it

would

be

desirable

to find an alternative to

the

_cycling

operation. This is _particularly desirable if

the

controlled

flow contains abrasive material.

Ideally,

from an equipment

standpoint, one would like to find a scheme _whereby the

motor would turn on in response to an error _signal,

drive

the

valve to

the

correct new _position, and then turn off.

However,

in addition to equipment _{considerations,} some means

for gauging the "goodness" of control must also be a part

of this scheme. In the analysis that

follows,

the

integral-square error is used as the performance index and provides

this "goodness"

measure. This index is minimized under the

constraint that _only one reversal of the motor will be al

lowed for a _step change in

input.

Therefore,

some form of

on-off type controller would provide the _necessary control

signal..

An on-off type controller can also be used with a process

requiring a continuous input

by

_adding an integrate and hold

circuit _{in the} controller output. This circuit has

the

same

output characteristics as a motor driven valve and

the

analysis of this paper applies to both systems.

However,

the physical phenomena

taking

place in the two systems are

different.

In the system with the motor, the sequence of

during

which AC power is

being

applied to the motor ter

minals. In the system with the Integrate and hold circuit

the pulse sequence is the variable period of time

during

which a constant input voltage is

being

applied to the in

tegrator.

In both systems the excitation will be applied

so that integration in the proper direction will occur.

The control problem becomes one of

deciding

how to synthe

size the pulse sequence to be fed into the motor or inte

grator circuit in order to achieve the desired plant

response within the constraints _{specified. The} analysis

in this paper develops a method for

doing

this and leads

to a system which is a departure from the current approach

GENERAL THEORY

A,

System Description

The block diagram of the system is shown in Figure

1.

It

consists of a pulse-width-modulator,

integrator,

and a

plant. The plant is a linear system and _may be described

in Laplace Transform notation

by

the transfer function

^

=

raW

where K is the plant gain.

The plant input signal will have the form shown on

the

diagram.

This signal is produced

by

integration of the

pulse-width-modulator output which is also _shown.

In the

following

analysis the modulator output is assumed

due to a _step change in the reference input R(S). The

function of

the

pulse modulator in this system is to cor

rectly synthesize the pulse sequence required to minimize

the performance index subject to the single reversal con

+>

3

H O

O

cfl

s

CO

rH fa

w

a

o

H O fa <j! M

o

CQ

6

B,

Derivation of Plant Response Equations

Let

the

normalized plant output be

C(t)

and represent the

dynamic response

by

the vector matrix differential equation

where: A is a constant system matrix

x(t) is the state vector

b is the control vector

u(t) is the plant input

t is the normalized time variable

Since we are only interested in the plant _output, the phase

variables will be used.

Therefore,

the state vector is

where

and

jr

(-6)

-

Kit

-a.

I/,

M

-frrt)

'Xj.

(6)

From these equations we get the

following

A

=

o

i

i -ab

~(a--(-l>)

1

C(-t)

=

[l

*]j

(4)

The

solution of equation

(2)

Is

to.y^&M

'fa^o*

'

The

state transition matrix can he found as follows

tl-l

&

66

Also

where:

A

s are the eigenvalues of A

n is

the

system order

The eigenvalues of A are the roots of the plant character

istic equation and will be assumed to be ordered as follows

From equation

(7)

e*= <*-,*

(8)

Therefore

-^ y

*FT-6

tt*

)

(9)

From equation

(6)

4-b

**=<<0x+,

(ye.

~*.&

J

>~Ub(^e)

-(e-^l

(10)

When initial conditions are given at t

tQ

rather than

at t =

0,

_equation

(10)

_becomes

&(--t)

j_

0

-*.(&--)

^/>tt-*)\

fe'fi'^l

*&-*<))

L

-a(>(<

-a(-6-*.)

-

*

fr-th

.

/

?**)

,

4*>tO

C.

Investigation of Plant Response

We will now use

the

general response equations developed

in

the

last section to investigate the plant response for

various

inputs.

The objective is to obtain insight into

the

problem of _selecting the best input signal to achieve

the optimum _operating conditions as outlined in the intro

duction.

Let us assume that the plant input is a _ramp followed

by

a constant value and compute the plant response.

/9-&t

06E**,

where k~/9 t.

The steady state output is from equation

(1)

If the _{desired steady} state output is _m, then from equation

(12)

From equations

(5)

and

(10)

_{for x(tQ)} t - 0 _and

u(t) -# t

%(t)

-b-a.

O

L

%,

y

(^y^}

(y^6^r)}

j

]Jr

.X,

-t

(t)

jLJ^-6

e~yt-r))rA

'r

d3)

Performing

the

integrations indicated in equations

(13)

and

(lk)

and _using the notation of equation

(3)

<*(*)

=

Ca.t>y

<*<>

+.16

-apt

(15)

M)

-_

jl

t

jl\4ZE

y*

(16)

The plant response for the constant input k can be found

from equations

(5)

and

(11)

X>jT

&-*

-h

L

^K

-m

<***>-

y(i<))

y-^^H!^

a

jf-a.

'tr./

_*.(-r) -hC*-*),

E6^-%<E"(i-rJl

C4t

(-e)

=

[*&)

+X,

(tP

-

A6]-6-*,)

(17)

<E^)

= -666h^+^}

J>K~\^(E-,)

a.

b-a.

L

M.1

^ft-*')

(18)

Figure 2 shows the plant response obtained from equations

(15)

and

(17)

where the constants _{have been} chosen as

a-l,

b-2,

K-2,

k-1,

and^

-

1.

The desired

steady

state output is

C(<x>)

-

1,

_Curve A _shows

the

_{plant re}

sponse when the input _ramp is stopped and held constant at the instant it reaches the level required to maintain

j | , 1 i ₁ l 1 1 | i _| 1 _|

'

i 1 I _| 1 1 ' ' _,_ |_

_y ii

j

: j 1 1 i !

'

L

1 i _{1 |} i i

| i _| [

f~f

- - - 1

1 i

"

i i i i _in 3-

y

4

-'

| ; j J __ii_) l_

;i I ! i

1 '

I6eF~ _l -1- 111 _J_II r*

H

1

1 ! iH . _E-^r3

*-o

j-ii 1 i _i +1 p-< * ' ;

'

~r co

\

| i

-o

4- --

r-t- -J- -t-

O

-4

6

1 i i i ,-.

1 ' '

^

Il

_,_...

i

i. -": ^ ^ <

-4- . .

1

6

\

A

h ~^t- 4 > 3T 3t

\ '4-

-\ T

1

-

1

o

1 U:

I

^ : ' !

-0

CO w

\

1

V

'

\

V

66

I

Er 3t333lL:

iz

-A 4-4- -1-

-h-\

EH

V

i o M

V , , . \ ! i - _?*?,

,1

\

\ "3"

H

\

1 Eh Oi

\1

\

₁

\ 1 \

\ . 1

CVJ

\ \ I ' ' ' ' ' _i

\

\

1 ' ' '

1 M

\ _\ ! 1 ! 1 ! _| ,i _K

N. v i'i . )

x _{\ |} '

'

!

'

i . i i

p~> r^

\ \ ; i '

j i , i | i

M

\ VI i . _i '

i ! _rt.

\ \ 'III

""N- ' _* '

*L x

^^">-

X 1

v X. !

'**^s 1 > | <->

'

''S~^->J

|S^v '

i**"1**! ^^ """LEX: 1 ^"*^jj ' 1 '

^""^"-v

"*^

^^s. 1

_L_ _,_

'

F

S4

j_ 0

^ . ^V ri 1 s J X-\ V '

|

Y 3 " 0

in 0 *n

0

,_, rH O

Curve B shows

the

plant response when the input _ramp is al

lowed to run on past

the

required _steady state

level.

This curve shows a considerable reduction in the error which

exists _up until

the

time

the plant output reaches the de

sired

level.

Obviously

_{the steady state} output is consider

ably greater than the desired _output, so that the reduction

in error in the lower part of the response curve is more

than offset

by

the increase in error in the tipper portion of the curve. It is _easy to see that Curve A with the as

sociated large error is the best response possible if the

input signal is restricted _{to ramp} and hold at the final

level.

The Improved response of the lower part of the curve could

be maintained if the input _ramp for Curve B is _switched,

at the appropriate

time,

from a positive to a negative slope and allowed to _{ramp back} and hold at the final re

quired

level.

The switching

time,

T , must be carefully

s

determined in order to prevent large overshoots _in the

plant response. Figure

3

shows an input of this

type.

From Figure

3

r

fa

0tTc

"M^Sfifri-*),

rr<-**(J*-f)

<w>

(rryf)6-3S^|^3^ffip333:|33333E33^

Z _I ZZZZZZZZZZ" "ZZZZ3 "ZZ" ZZZZZZZ3

1 1 1 1 I I I 1 _{I 1} . 1 111

1 I I 1

|

Ml

|

|

u u u i

zzzzzzzzz:zzzzz-=============

=========zzzzzz:zzzzzzzzz:

j

i i i _{3 3i_ZZ_i}

| " i ~ | _ ZZ" " 3ZZZZZZZZ zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz^zzppzzzzzzzzi::: zp_

|

[

|

[JZ-L-^iT-u

666\

1

!

!

SF^rr~Hi"- :zzzz ~zz

EEEEEE^E^ffjff

1

\\\\\

_\i

\

\

^ffi^^^^===========

_ _

| |V| I

U^"sj

|

-So-I ra En CM to Eh M EH EH Ph Eh

3

Oi Eh PL, o M C5

The

plant responses for this input will be defined as

follows

4lJ-,

/

JUL)

(20)

C^(t). and

C.,(t)

are given

by

equations

(15)

and

(16)

respectively.

C2(Ts,t)

and Cg(T ,t) can be found from equations

(5)

and

(11).

-^C/CrJ+cXvh

*fSrO+"*Jm

-*(--!}) e,

-b

(-/<.)

(21)

*rfo)

-

ir^

+ZEiWiC6)+6

(r6-p-(iH-6]

-*&-/z)

Mt-ix?)

e

C_(m,T_t)

and C,j(m,T ,t) are also found from equations

j s J 8

(5)

and

(11).

b

J

Jr

(23)

iM^6\^y?F^y-P^6^f)

6zhkyr-p4ky-f)->fY(^P)

2

D.

Formulation of Integral-Square Error

Consideration of the response curves shown in Figure 2 led

to the heuristic conclusion that an input such as that shown

in Figure

3

would _appreciably reduce the response error. As stated _earlier, if

the

switching time is too large the re

sponse will exhibit a large overshoot.

Obviously,

the over

shoot can be controlled

by

_controlling

T_.

Smaller values of T reduce overshoot and negative error, but allow the

S

positive error in the lower part of the response curve to

increase.

The undesirability of an error is Independent of its alge

braic sign and we are _primarily concerned with error magni

tude.

The larger the error, the more undesirable it

becomes.

The integral-square error is a performance index that

heavily

weights large errors of either sign. It will therefore serve

our purposes _well, and the optimum value of T will be found

s

based on the ISE criterion.

The plant input signal is a _ramp and some

finite

amount of

time is required for it to reach the level which will sus

tain the steady state output we want. In addition to

this,

the response of the plant to the applied control signal is

application of an input signal to the system, errors must be

accepted as

inevitable.

In any given _system, the ramp rate A is fixed and determines

the length of time required for a given input level to be

reached. Until this level is _{reached, nothing} can be done

to improve the plant _response.

However,

from this point on

in

time,

the system designer can control the. magnitude of

the

error through the proper choice of

T_.

Therefore,

only s

the error in the interval -jr-- - - eo is considered. The P

error will be defined as

follows.

6)='^e^(^Jrrj-t)jr^^(^-f)

(25)

The ISE is then

iz^ld

J6?~

fi

oO

'jih^y)^

+}*

We could procede

formally

to make the substitutions for

the error _expressions, do the

integrations,

differentiate

the result with respect to T_ , and solve for

T

by

set-ting

-cp--0. This is a very formidable task and it is

desirable to reduce the labor as much as possible. In

spection of the first integral in equation

(26)

shows that

the integrand is not a function of

Tg

and further that

Tg

appears _only in the upper limit of this integral.

If the integration is performed and the result evaluated

at the

limits,

only the evaluation at the upper limit yields

an expression

involving

T The next _step in the procedure

5

is differentiation with respect to

T_.

As a result of

dif-ferentiation,

the integral evaluated at the lower limit

vanishes,

We know that the error at the end of one response interval

is identical to the error at the

beginning

of the

following

response interval.

e,

fas)

=

e.

fat,

J

77)

(27)

This means that the first integral evaluated at the upper

limit is equal in magnitude to

the

second Integral in equa

tion

(26)

evaluated at its lower

limit,

but the algebraic

signs of the two terms are opposite. These terms will can

cel and

therefore,

the entire first integral in equation

(26)

ultimately vanishes,

It can be shown^that

by doing

the differentiation

first,

the terms which will _eventually cancel will be placed in

evidence at the

beginning

and can therefore be eliminated.

Also,

the terms not

involving

T will vanish.

Therefore,

s

In order to insure that the first integral vanishes at the

outset, we will change the normal order of operations and

take the -w^

first.

d is

Let:

77

1

<r

%.to**>r:)-tftftop+Jfr!M*

fri

fi

=

Let:

Zr

^76

--4^-f)4C^/z-f)-^(-r6eU^r6

-J/2-fi

+)iFs46K-t)J

T<r

-3&.

fabi1*-?

-f)

-

4(^t^

-^J-k^c^s6^

7?

(29)

Let:

jrr,~#-&7j

y

^

/

__

JUL

J7F-fi

=

-i4^~SjATyf)

+J-k46,rA^

(30)

By

combining equations

(28),

(29),

and

(30),

and _making use

of equation

(27)

J7}

* * 7s

/7

J

'

(3D

is

fi

The first integral has vanished as _anticipated. Equation

(30)

can now be set equal to zero and solved for

T_.

s

7~-J/t~ *

fy*(nv>*

y&*w*

=",

JT--J2.

E,

Derivation of the

Switching

Equation

Refer to the integrands in equation

(32).

=

-1*1*4

c*y)

*

jt?

<4fcy)

(33

Similarly

<&&)

--*r7z%

(&)

+**<(>>**)

(3k)

Substituting

equations

(33)

and

(3*0

into equation

(32)

and _changing the order of the terms yields

-*)&,

fc

*)

y*

-a^j-fe

dj

fa-u,*)

a

^

fi

. co

+J-k4'(-&)*

+$i*4

(,%*)*

(35)

Consider the first two integrals in equation

(35).

CQ(m,T ,t) can be expressed in terms of C2(T ,

t),

plus some additional .

terms.

See appendix. If this is

done,

it can be shown that

these integrals reduce to

*66o-ir^&f6

Equation

(36)

shows that a _particularly simple result is

obtained if the _ramp rate

A

Is unity. In any case,

the

first two integrals reduce to

(36)

^LiL

-

*MAJL

T

(37)

/2*lo /ZLfo

r

Evaluation of the remaining two integrals in equation

(35)

yields equations _containing both algebraic and

transcenden-2

tal

terms.

The results _for the integral

Involving

C0(T0,t)

c. s

are given in Tables 1 and

2,

Table

3

gives the results for

2,

the integral

involving

C,(m,T ,t). The entries in these tables are the coefficients of the terms appearing at the

left of each row. The numbers at the

top

of each column indicates the term in the derivation from which those par

ticular coefficients _came. As will be seen

later,

it is

o

H

I

I

O

o

o

o

ty)

S

o

o

o

O

^> j in o

\

t

*

^

o

ty

MS

o

o

O

o

o

o

o

^3-o

^

H*

-I

O

o

o

o

o

o

o

o

o

w Eh tn EH O En

wv

>

^

k-1

'ty

0)

^

^

ty

1^7

7

w.

^ S

^

1

V

*>

I

as

o

0

o

o

O

o

I

i

4-o

O

o

CO o

o

o

-S>

o

o

u

o

O

ty

1

^3

i-i

t

-f-^

r

*.

^

o

4-o

o

i

i ty

o

o

o

O

O

O

cc w EH tn EH 1 53 OEh o tn

^

\E

k>

'ty

'ty

V

ty

>

'ty

several expressions which make _up

the

coefficients of each

term,

rather than combining them into one

long

expression,

All of the letters _appearing in

the

table entries which have

not _previously been

defined,

are defined in the appendix.

The final result from carrying out _the indicated operations

in equation

(35)

is found

by

_combining the coefficients of

like terms In Tables

1,

2.

and

3,

and _adding -to

this,

equa

tion

(37).

We now have an equation with _only one _unknown,

T_.

Solution of this equation yields the optimum value for

s

T

The switching time equation is most _easily solved

by

_using

a digital computer. A general program would require that

all the symbols used in Tables

1,

2,

and

3.

be expressed

by

their definitions in terms of the plant and system varia

bles.

Then,

a statement must be included

in

the program for

evaluation of each of these

terms.

This is _generally not

worth the effort unless a

lengthy

analysis _{to determine the}

effect of variations in all the parameters is to be conducted,

A good procedure to follow is to construct tables similar to

Tables

1,

2,

and

3,

and use a desk calculator to evaluate

each of the entries in the coefficient

tables.

As each ex

tables.

These results are _easily summarized to yield the

switching equation coefficients,

A simple program such as indicated

by

the flow chart in

Figure

k

is sufficient to find

T_.

The operator must

pro-vide the limits or the time interval to be searched for a

solution, the time

increment,

and the input step magnitude.

The program evaluates the coefficients of each term in

the

equation.

Next,

the equation is evaluated for values of

time in the search range. The print out is the term co

efficients, the

time,

and the equation value for each

time

value. It is not _necessary to print out the coefficients,

but this information is helpful in

finding

errors.

If the solution for T is in the range _searched, the value s

of the equation will change sign between two successive

values of

time.

Initially,

it is suggested that rather

course time increments be used.

Succeeding

print outs will

indicate narrower and narrower ranges _{for the} solution and

the time increments can be reduced _accordingly until T is s

INPUT

Tg

(max,

min)

AT0

m

CALCULATE

COEFFICIENTS

PRINT

COEFFICIENTS

DO

3

J=^min,

TSmax

EVALUATE

SWITCHING EQN.

PRINT

RESULT & t

END

FIGURE

k

EX AMPLE PROBLEM

A.

Objective and Plant Description

In order to illustrate the formulation and solution of the

switching equation, a specific example will be worked. Con

sider the second order plant described

by

equation

(1).

For

this example the plant gain is chosen as

K-2,

and the poles

as S. =-l,-2. The plant input signal will be considered a

unit ramp.

B. Calculation of T s

Using

the values given above in equation

(37)

we obtain

***-

4-

mX-(38)

Evaluation of the coefficients in Tables

1,

2,

and

3,

are

given in Tables

k,

5.

and

6.

Summarizing

the entries in

Tables

k

and

5,

we get

ti-66

j*-

&<?)***+

[o-^-*-)*]*-*

+(&*+,

^

ty

4.

o

o

o

o

*1*

J

!

ty

o

o

o

^

'

ty

*K

^>

^s

4*

<

%

)

o

ty

o

o

o

o

o

o

o

Tt

^

<^

1

i

M)

*k

O

o

o

o

o

o

o

o

o

CC

w

EH

1^

n

^

i

ty

ty

'ty

>

₁

ty

k^

ty

'ty

1

ty

*>

Os

U

+

o

^>

o

o

o

1

1

ty

1

ty

o

c>

Q

OO O

1

0

o

^k

c^

o

*-

4-o

o

ty

r

ty

i

t

ty

o

o

1 J

r

ty

O

^

o

o

O

o

s cc w EH tn Eh 1 s O &H o tn V^5 kS 1

ty

?>

ty

'ty

ty

'ty

ks

v>

tn EH -H t-Q

1

^

o

I

^

ty

ty

)

o

4

ty

5

4-ty

o Ph W o o o H EH of w o i-H Q

4*

k

Q>

C^)

o

o

o

ty

1

1

>

1

ty

<^>

^

O

g

M X u EH M 3= OO Ph o o H Eh Q . >k

o

o

5.

ty

1

^

^>

o

o

<0

^)

3

1-3

W

1

VO

8

o

o

<^

o

o

*

^

^

9

<^

o

^>

O

o

Eh Q

k

o

^

1

^

c^

o

o

o

o

₀

<o

5h 00 &i O h o tn K17 k^>

"ty

k^

ty

k^

ty

ty

*k^

ty

J

ty

ty

k?

'ty

fc?

>

Collectlon

of coefficients in

Table

6

yields

-(ye

)e

'+(Te

)e

+(&

+T&

~T&

)e

_

(ff+

uT)e*+

[t*6

(? +,)*'"]

(gM+~f)^My

f

h^

+*f

-376

e

(ko)

The

switching equation is found

by

_combining equations

(38),

(39).

and

(40).

i-i~-K

+

(^-e"+--)e~r

+26

-4*171

=

o

(*D

This equation can be solved for the optimum values of T for s

various _step input magnitudes

by

use of the computer program

previously described. Table

7

lists the optimum

T_

values

s

TABLE

7

OPTIMUM SWITCHING

TIME

Step

Amplitude

Switching

Time

.0125 .141

.025 .201

.050 .290

.10 .423

.25 .711

..50

- 1.081

.75 1.^01

1.0

1,697

1.25

1.980

I.50

2.253

J 1

tz

6

i

i i i

i_ !_ <

i i i i i

'

i _[

I |

V ! .

.-\( l

\ i

\| |_

ZL... _\T ~^

\

t

zt

\

\

Is. i

\

Z

3ZlZ

V

zr

S3 ZlZ _J_ Zt Zt

\

X

zt |Z

\

zt

$

zt

\

\

\ \ \ \ ' \ \ v. \ A

-y

X

Zt

^

Ar

-+--4- -4- Ay-

-h-V

1 1

\ \

1

\

: I 1 _\ ' _! '

I ! Iii i

! \l

-4-Z3Z. Zf3 3V Z3Z .ZiZZt

-h-

6p

zt zt v zt i

zt

n~ ^ 6

^\

\ \ \ "" "

i i i \

i *,

-L- ZT ZtZK Zt Zfl

- -\

\

._ "' ... \ \ .... -6 v ^ ^ \ . \ ... !\ ... \| \

T

\

X

T

4^, i\

3*4^

...

-!.., ^*- M

o CO vo EH M CM o w J Eh ^ CO H En O M o EH M 3= 00 s M EH Ph O W

g

O M fc VA O* o CA CVI o CVJ vn o r-H VA O

Plant response for the optimum _{switching time} and for switch

ing

times greater than and less _{than the} optimum are shown in

Figure

6

for a unit _step

input.

Clearly,

the best response is obtained when the plant input is switched at t -

Tg0.

Study

_of

the optimum response curve shows that peak overshoot has been

held to less than

3.5^.

The straight line plots in Figure

6

are the plant inputs _{corresponding} to the three response

curves.

Figure

7

shows the optimum plant response for several input

magnitudes. In each _case, the desired _steady state output is equal in magnitude to the

input.

The ISE can be computed from the error equations

by

use of

the trapezoidal rule.

(e

-[m-c

(t)J,n

-

1,2,3;

_{refer to}

equation

(20))

Evaluation of the ISE

by

this method requires

only a simple computer program. Table

8

gives the ISE versus

switching time for step inputs of

0.5

and

1,

The table shows

that the minimum ISE does indeed occur at the value of T

s predicted

by

the _switching equation

(Refer

to Table

7

for

T_

optimum). Figure

8

is a plot of Table

8 data.

s

The ISE versus T plots in Figure

8

indicate that the ISE

increases much more rapidly for switching times greater than

i \ H y-_, L J_

I ' '

... m

~|

i i

i

6~

776

r Zl3

r- 31 r i c~ ... " 776 Zt

zt

t

zt

zt

J3

ZZ

i

Zt 7 Zt

ZZ

I

/

.

~h~^~

6676E

EL 76 Zt Zt

^ _yp-6

tt

li

/

/

_lia_JL _u.l

_u _l_ i

TrpFl M -+- -t~

t>

l' , 1

_^i_Sj__iL_iL_ _i_ _l_ ^_

A _v

\

_i

/ ' l<s '

1

\

6

-t

/ ^: ..'.

ll,r

i*

T

ErF\

\

FT

kJ

|\ r

/ *>M'J !'V| lit.

'

\

k?

/

\

l1'/--4- 4- -4v-

-h--r

-f- +- -*ft^- -+-

-+-\ V \ 1 fA

\ (t "^

\

\

-*o

\

V\ \

\

^S>-*J \ |\

1 """"-"J. N

'

\l

| i _1_-r^"^-^ \ 1

\

_^

r~*~"| ^"-kn^ xl

1 i~~>~-v^' ^v !

i ^""^i^ _^kj

- "-'^Uo^f ^"^k. "^**"*_

^^****-l

isT**--~~-_ _j_ ~^^ i_

""""- ~^

"

"""--Zt

""

""^

"^-" ^ _^

~"^-- _,_ ____ 2k

.^

+

-=^ -p^ .^ "s ---.. \ "j^. V

^L '

"k^ V

----.^ j

^~-==.

1 1 1 1 , 1 1 1 II 1 1 1 1 1 II

ITkko

h-i EH VA VA CVJ VA O VA CVJ

'

1 1 1 1 1 1 1 TT 1 .ill _{| | p}

-r-i 1 ₁ "

...

.

-00

1

...

I''-""' . , , .

^

1 1

OT

Z 3 _' Ztzt P_i>

--

r-& E>

"J O

-. * 1 Z " Zl3

g

S o Cu

4 y- '-'- - 6P _r?

4 - - _ _J_ _j_

7

-i _^ M _y

Zt

J -j Zt _i_ ..it _^Z^Zl H

B

4 4Z _l.

2

4 3ZZ ..

EH

1 cu

O

l\ :

V

3

zL

_t

zt

zt

_. ..

\

1

\ l nv n

\ \ 1 w

V 03

V. \ ^

v^ \ V

-o

zt

^^

%

3 ::

-

ss

_^

ty

4-+

~~*~>is- v, \

^s

k

^^^

^

-f-^^vl **

\

^""^.^ \. \

2p=ik V

a>>*k

\i ^k. \ ^<^ V ^vj \ ^\ " " kk ri

'V

^^ *s \

s

V 3JZ

\

i

lp, vr\ O vn o in o

cvj cv vn oj .

H rH .

-. . _q

TABLE 8

INTEGRAL-SQUARE

ERROR

Input

Step

- 1

T

s ISE

I.25

.447

1.55

.375

1.65

.363

1.697

.361

1.75

.363

1.85

.382

2.0 .454

2.125

.564

Input

Step

-

0.5

!s

ISE

.5 .169

.75 .148 .90 .128

1.081 .115

1.20 .123

1.40 .192

ll I 1 1 1 1 I I i i i i i _{i i i i i i i i i i i i i i i i} i rn i i i i i i i i i i i i i * . ^^ ^^.^ V ^N,,

.rfS^ , _o

ZZ3 ZSL

^.r

.

-A/ . . >!. f\l

E^ |

I _N^

V-\ in

V

k

-T-^ _, ,_ _,_

Al_'-r-' Zt"

Zt V .

\

T H

}.. . ., _ __ 1 .... ,_. _ T.m

1 '

J

r-7

Zt

EJL^

/

""k-t

P6

^kZ ^

r^

'/ "N

-/

ik

/-

-$6>-^

/

4^

-

M-*. \/\ ''. '

N,

I 4 N,

/ \

7 _s; vn

/ i \ cvj

y \ \ *-\ \ \ t-X

._ ._ .- C .. .... _ ... ,

T

-4-H ^ zt i

tt

^

3k+

zt

X

t

16

L-,

-

^

t

r^

4P

-J-i-l--i,,l,i.l.., , III

caused

by

the rapid increase in overshoot for larger

values of

Tg,

Figures

9

and 10 are plots of the optimum

pulse-width-modulator output and the _resulting integrator output for

^1 43 fc Eh !=> O fc

1.5

1.0 Eh P-i M

50-5

0.0 I T i | i 1 . ... | _^ . _ -y^

1 1 1

' "* _i

_| 1 1 1

zt

1 i i 1 ' , /l\.

2 1 >.

/ S

Z

/ V.

z V. 2

^^.

2 s 2 V Z

^v

/T N.

,/l X _j_

p jf / ~7 ~? 2 2 ^ ^ ^ I^Q&I L.ii7"

Z it

I

_ .

1 2

3

TIME

FIGURE

9

- P. W. MODULATOR OUTPUT &

PLANT

INPUT

i_ I. i

'

i

i

u Zt Zl

| ;. I ; I

'ill!

,

f~

Eh

t

u Zt

3

!

fc + I ,

B^

3 O

,n I ! _

a

="-|

-- " ' ' ' ~

j

fc " "

'y 't r\ r..'J .a

Y

6

/

**

1 r-

y

\

/

\

\^

^

v

"~> / Sy

z ^

^ ^

r> j/ ^

Cij 1 n ... S , \

5 l.U

-7

ri x

^ A

5

z fc <^-n r-. /* U.J

/

2 ^ 2 2 2 ^

^ 1+166 ^pij.e

2 n n Z

0 1 2

3

TIME

FIGURE 10 - P. W. MODULATOR OUTPUT & PLANT INPUT

CONCLUSION

This paper has presented a method for

finding

the optimum

switching time to minimize the integral-square error for

linear second order plants

having

real distinct roots. Cer

tain types of industrial control systems have fixed elements

in the form of an

integrator,

and an element of this type

has been included in the analysis.

The data in Tables

1,

2,

and

3,

together with equation

(37),

provide the necessary Information to formulate the switching

equation. This equation is general and can be used to in

vestigate the effect on switching time of changes in system variables such _as, integration rate and input _{step amplitude,}

as well as changes in plant gain and poles.

The pulse-width-modulator in the system discussed in this

paper is required to respond in the optimum fashion for _any

input amplitude. In

designing

the modulator, it is

therefore

necessary to know the relationship between T_ and the input

step amplitude. This design information is provided

by

the

switching equation.

tolerated,

or at _{the very}

least,

is

highly

undesirable. A

worthwhile extension of this analysis would be to rederive

the _switching equation to minimize the ISE under the

REFERENCES

1.

Schultz,

D.

G. , and

Melsa,

J.

L. , State Functions and

Linear Control Systems , McGraw-Hill Book

Company,

New

York,

1967

2.

Newton,

G. C, ,

Gould,

L.

A.,

and

Kaiser,

J. F. ,

Analytical Design of Linear Feedback Controls.

Wiley,

New

York,

1957

3.

Buck,

R.

C,,

and

Buck,

E. F. _, Advanced Calculus.

APPENDIX

A.

Derivation

of

C3(m,Ts,t)

_{in terms of C2(T}

,t)

From equation

(23)

-ttfa

&*&

+4

k

*-t)-*py

***?>

+

a.b

(Al)

The C2(Tg,2Ts-^)and

C2(TS,2TS-^)

terms in equation

(Al)

will

be expressed in terms of

(^(Tg)

and

^(T

).

_{Refer to equations}

(21)

and

(22)

^y^rJ^/T^^O^^)]^^?'^

&--&te-f-*-Zr)

'

m[

*-(*-*+

^

\c,fc)

+6(V)-^

O+'Tr)]**^^

+

K

>niE

_

j

+l

+

M

(m

)\

J-"

&&*-?)

HE

At, a.

'

<*

\

at-/]

& r

-M.

b

e,

(t?)

+

C,(rf)

--%r

(t**t)

e

+

B.yf-iy]e^-^P

-"/*-%)

<A2)

^

t

*r

L\..

J

/_ M \

h*Jlk~\

~*(-6 ^'s*_#

J

a.

C,

0Sj

*Q-j)

1-t{%Jrrt-f)--f-\*

f

-16

i,

b

rt

4 (Tj

^4

fe)

-

M-

(f+

&)]*

-b(t-K)

f-fintk

_

fik

MK

fik

_{/(o+< )1 -bl-t'l's+T)}

.

z~

*,b

h

b(irJJe

r

a

clfrJ*cl(T6-g(l^vj\e,'(*-v)

+

6

lh

~

f

t,t-*1yt>(--J6TJ-)

M

**,&)+*,

ft)-

f-O-ri&l*

rb

(-6

-ti)

Jr

Jl-r

h

(A3)

Substituting

equations

(A2)

and

(A3)

into equation

(Al)

-f-+

b(y-*-)

hn$k

/Z-b

(A4)

Comparison of equation

(A4)

with equation

(21)

shows that

the

first two terms to the right of the equal sign in

equation

(A4)

are equal to C.

(T

,t

)

?

(t-2T

-~

)

c. S ^t S ^C-b

l6\e-*(P-Jp-+f-)

+&>[(t-0+\

h

yisk

a.

b

B. Evaluation of

EL

6

P

o^

r- ^Tr^

Consider the first two integrals in equation

(35)

and replace

C_(m,T ,

t)

in the second Integral

by

equation

(A5).

3 s

OC?

^J^faJJk -J^Jjz^fa^J^

<&y-&y

fs

*v

-Mr

-3.ME

fik

\ji(b-^)l

tir-O**

W

(.

p

d

e

P

)

+JLM

k

\b(h-*)\_

(t")+t

oo

-Jz*

b+*\

.

f"fik

y-e

JT?

<&b

Combining

terms and

simplifying

_{in equation}

(Bl)

yields

-*)&>&.*)

J*

Tc

cfi>

~^/v7

\6t-or]

\

!

i

~M

1

y-t

OO

-h^M

(B2)

Notice that the upper limit on all

integrals,

including

the

one

involving

C(T_,t),

is now Infinity. As a _result,

inte-c. S .

grands

involving

terms of the form _& ' _{will vanish at}

the

upper

limit.

Most of the integrands do contain terms of

this

form.

Also,

it should be noted that the lower limit on

(i.e,

constant with respect to

the

variable of

integration)

Therefore,

evaluation of the integral at the lower limit

yields a constant which is just the positive reciprocal of

the coefficient of

the

exponent. Since we _previously estab

lished that the integral vanishes at the upper

limit,

this

constant is the value of the

integral.

For example:

rft-p)

. i

(B3)

In

the

analysis that

follows,

we will

frequently

encounter

integrals of the form of equation

(B3).

The above observa

tions allow one to evaluate these integrals

by

inspection.

In order to complete the evaluation of equation

(B2).

C(T ,

t)

y~ s

must be expressed in terms of the system and plant _parameters. This can be done

by following

the same procedure used in

Refer to equation

(21)

ye,CT,)

+

i,(r,)-#(,+*v)y*('~7:J

-*6

,

M./-7-.-^)

+--j6

,**_

_

jK_

\

-*?

M.

fc-4$)

0*

/

zr(/+*v)

or

-*Hr

+

k

a>

L

jt

+

b

h

J7

4.L

b

L(kzS[e^

+

M

O^

LAk

_\. tb

J

)

&

_* "*

M6

-*(*-r6

^'

<5

M

OE

(i-a<z

)e

(B4)

[**l&j+'tM-$-o+n)y(*~vJ

=ee16^_i6.

\&-r*

,

I6(r.^y\

+

?A-*i

6(b--)>&

+

b

(^

ir

]

+

Ml

a

-j~0^)]^^6

b-a.

U

bX

J

f

16

6

qrH

J_

_i

-r

I

rb

&-%)

ab

+

"

b

S

6

yb(t-T,)

*

e

60-^ir')e-li

(B5)

Substituting

equations

(B4)

and

(B5)

_{into equation} (20)- we

obtain

-

M

$(%*>

=

z&

o-^>*

0-T<r I -& *

^-

(t-xe^)e-^

*Ywj

^6

L

^^

J

Taking

the partial

derivative

yields

^<v;

=

j?^

-*

^

#-",/

*-(b-*)

(B6)

-f

UL-*-*>

(*-T?)

b(b-*>J

Using

equation

(B7)

we get for

the

first integral in

equation

(B2)

i

-2m

IjpCjT^)

Ji

ti

-

*+'&,(>*>*

p.

it-*))

e

&

Tf

fa

-4?- -fa^ -^JJL/'l

Integratlng

the

_remaining terms in equation

(B2)

yields

'"hfebtf-'^l

+4-dzl"tt-'>*rt.

+

fl

(B9)

27^-J*~

P

Notice that four of the six integrals involved in

finding

equations

(B8)

and

(B9)

are in the form of equation

(B3).

Combining

the results of equations

(B8)

and

(B9)

and simpli

fying

we get

'_-

-

J

-

jha6-

t

1

-1

)]

^

[*Y*-*J

aHb-*)

&(h-*)

lP

J\

fEw

p

_ _6

+

EpyP-

(-i--i)

+

+.

//)

ab

*rf-oc.

fr-Af

}J*

-j

Ji

-)A

b-*

3-76 -H

*"6j

-f*

TZ

X

^&JU-i

b-*--+(**-?-*)

0~b r

a.ip

4/^fik

^h

"^

/t\> a.b

T*

Therefore

XT'?

ti

iV-Vf

;

"

(B11)

C.

Evaluation

of

c?<r

From equation

(B6)

rif.16

%fc*J

=

(>-**

')e

-a.*

PK

/,

~h*

LA(6

O-b

-t-1%-b+t*'

cvb J

(Cl)

After _squaring this _expression, the next operation is differ

entiation with respect to T . Before squaring equation

(Cl)

5

it is convenient to redefine the coefficients so that the

terms _containing T are grouped together

following

the squar-s

ing

operation.

-4& _-b*

(C2)

and

where:

From

these definitions

the coefficients in equation

(C3)

become

aZ7i.

aAC .

flS,^4Lr(j(>+f7E)(i-3e

')

$flc

=

jn

(4>+g6)

(/-^yV)

&>=

fO-AerT'f

38c

=

j

rn

(r<i^r6

('-je

v)

d^yo^yT6

Performing

the

differentiation

we get

r<-.^77

=

j

rU-*(*f+py*r'-zaf7'

^^

ttz-

47?

jF

(ane)

=

4

n

(s

-z&f><z -.ap?

-^

V

e^)

37?

(&)=-****(!-*

By

comparison with

the

definitions for

A,

B,

and C

t'-S-ifiSi'*/'**/"

where ^

b^^

cz-b

a

jfik^

r-

A

**((,-*)

n

-~

h

(b-tz)

The definition of A shows that this term involves _p and in

turn the definition of _p shows that this term involves t .

These terms must be expanded before the integrations are

done.

Integration of equation

(C3)

results in some _very

lengthy

but

otherwise straightforward algebra.

Only

enough of the alge

braic steps to serve as check points have been retained in

Carrying

out the integration of equation

(C3)

yields the

following

results.

T<

rs

rr

-kfl^-ff-6*]

, El JL 4

J-M

/

)

^T$r*+-fi

I

<***)***

A

Er

_AY

/^~fi

7

6

J"

+3Lf(*feAV)

^e-^J-6

Tf

M_

f<*U

=.

-Jire

(i-ae*

^-Z^Te

Xr

/EEL

a.

-f

^n^lf*^-^

/z6Tg <z

Tt

-OEh7

X%-"/fi

[_

O^

If

77

, ..^

^uj^-^-<ay\

=

-!&(<-(^)^-^V^

-3?*>

??-6L(fiJ

fi

<ZL<M +i)e/8

e

-J&V

y^

-fa?+'J

-6

=

iIte-*H-jr,^-Af

-^^-Mfsfg

r*Hy

I

"-a^p*

+3.VJ6,

-hjrf/a.

-m.^Ty^

ay/L

fi

-0676-<&

(C5)

JL&-fi

-y-t

T-J-(3fic)&

V~A

jr*

f

tt

/-r-

J*1*

')je-'*S6

IPs

Ts

r

V

= -6M.fr-^_b-syr6(efie

~JbTs -b^r

;

~t-bnf<

'/

EP6r-f

Lb:

+3.hgjo/

(e.fie

4

= _

^!$-(jr.(,^)e.^-b^y'")(a-r<s

.bTA/

kp--MV

^

-<?

y

(-&

-2J,Tr &?

&

b^

fabV-js

+i

)-e

-bTr/,^

V

(*&')

bV

--iE$(^T--(b^)-2b%)

-^

[y^v-

(i,M-*)y*-*

^~hv]<sf

f

\e

-hi?

-mjr-e-**

-_

0%

<

u

.-"

fi

"*

tzPH. ,

^-Tr

= -M'h-a*')*-Je^A

(r

=

6ir

'O^js*^*-**-***)

-*y

(,-*<?*)***

**

-**7F

-ha^O-*^)* <

&

-4

V

&

+

4-?*

&V

U^P^y*

J

tt

T-b+*

L

*T<

,

bT

*e6iy-z(b*6e

l/l/l

tb+*-)TA\

Q**ty

-*((**.)?

e

'e

-e

b+*~

L

J

t/L

f

^

**

I

-a.l

-*rri&yi*-</-rftiIJze

-Fern

fat

b-t*>

M

pi

-gfn

<e

(b+*Ji$

-(b+*)-%

"j_

y^*ji

yu

(4--^4

(-AH*F-V-)

ZLK

e,

pA

-^fi

e,

r

^

'

o26J^ -jlcPf

/

&

-h

T^

(09)

Summation of equations

(C4)

thru

(C9)

completes the evaluation

of the integral. This summation is given in Tables 1 and 2 in

Oo

D.

Evaluation

of

From equation

(A4)

_2r

C?

(*j7Z,-6j

?7}~J6L

fi

M

k+d.

+

fiK

\

/-L--lu-LXEa-(*-z77+y)

_

J6E

yf-o+t\e

+

From equation

(B6)

-fik

/

J5i

-^

#

^

e

-;^(-^tfn^(^-2rJ

(Dl)

Substituting

equation

(B6)

into equation

(A4)

and _using equa tion

(Dl)

we can write

fi*

(b~"J

+

Mfik

(D2)

^z:

(It

will be shown later that there is an advantage to this form for

C_)

For the same reasons given in

the

last section

following

equation

(Cl),

it is advantageous to write the coefficients

of equation

(D2)

in a different form.

By

doing

a slight

amount of simplification in the _{coefficients,}

the

following

ryf6(t-^FA^r^6]

r

4

J*lfL

r

-.

f

J

-6-3-6+

j-(D3)

and

Using

the definition

for ll> in equation

(D4)

jzcifam)

-'fr-J-V+f-)

J-

/

.i-*6-3t+J6)

+-E-A

^yy^-^fKM^)^'mtf)

4(b+*JD&

r

+~3=q@)

e

fi

y

(D5)

The advantage of the form used for equation

(D2)

is now _ap

parent. When equation

(D5)

is

integrated,

all of the inte

grals are of the form of equation

(B3)

and the integration

00

/.

J

&

4

fay)

J*

s

fi

-

<//>r

i-7

fc

of)

i

j

+

?

+J-65-

(^f)

-* -J-

6.

ic*\

+^t

+JJ,

Jv

I*

/

^/D

+^i

(*0F)

(d6)

At this point it is convenient to redefine

D,

E,

and

F,

in

order to separate those terms which contain T from

those

s

which do not.

D

4

J+f

By

comparison with the original

definitions

we see that

j

M

ril(b-tZj

{*/->)

i-S.

0

A

fi^

/

f

=

7E6766 0-*e

)(e

a.-Pi-"-)

tQ,

,

6-(jt-26)

)

*

~

-,t)

(*")

(*

b(7~-2Zr)

(/

)

-abT -*>%

=

W

(&

-ze

J

f~*b

j

V~

afi(^)

C

J

W

-fa)

b

re

fi

The only terms containing T are _g and

j

.

Having

identi-s

fled these

terms,

we _may now proceed with the reduction of

equation

(D6).

, For the sake of

brevity,

as in

the

last

-section, some of the lengthier algebraic _{maneuvering has}

DF*f(y+f)

fifi

+

_a

~k

(SDF)

34-\

a.

\2*J+b*J+r*

<) ? . ,^//

z'

-**7r

~aT-2>Ts

/ _{-3L#-Ir -an* I}

J

/ _ r) .,!>//) -clL

(D7)

\4-MF+6,^(3DF)

=

tf*

(

J-

y

**'*)

F

-7

/

(J*f)

This expression is of the same form as _DF.

By

analogy with

the

previous derivation we _may write _,

sff

+y

jps

(jiff)

=*u-^y^)

y

=

(jt^y

=

y63fy+?*

j

See the derivation of equation

(D7)

for

/

<r

)

JT

KJ

o

*

U.J.

*>?+&

(*?s+F)

=

1^-

{Jv e

~*

6

(a A

Wje*-J.

V*e***

(D9)

EX=

(J>+jfi

This expression is of the same form as D .

By

analogy with

the previous derivation we _may write

I

J-

_J

4*+li

jk(

J

=

Jsft~6ylwe

6^.

+

fyJiA/+'Sw*)&

/-

/

...A

-367r

dl

=

(*fr$)

u+i

)

-

Js+jJ

+f-s

+jj

++k

A

(&*)

=

WiTi-tk^

(ft+jS+jt)

=

+0F

+

J.

6+a

^TrJP+^W+Tkap}

J&&--

= -

3a.E7i

v

u-^u-

.

yAr6

yk

QJ

-at,yw

(e^-*-**)

&

O'fJ

t/J>E^

</jS++s&\f

(e^-Ae**)

+sJj

C^^-^-^)

+f/Mv(6*(*+"jTFa6(H+*)7;-je-^>b)T*

6.

*Jtr

+

(*'*)

Jb+&-\

-o6t:

=

4U++M

(&)***<*<

(t)'

-**>

(W)***

+

j>*

te)

-(b+a.)

^

Summation

of equations

(D7)

_thru

(Dll)

completes the evalua

tion

of the

integral.

This summation is given in Table

3

in