FREQUENCY ANALYSIS OF SINGLE-VALUED THE · 2016-06-19 · DUDLEYKNOXLIBRARY NAVALPOSTGRADUATESCHOOL...

NPS ARCHIVE1969.06DAVIS, J.

FREQUENCY RESPONSE ANALYSIS AND DESIGNOF

SINGLE-VALUED NONLINEAR SYSTEMSUSING THE PARAMETER PLANE

by

John Philip Davis

DUDLEY KNOX LIBRARYNAVAL POSTGRADUATE SCHOOLMONTEREY, CA 93943-5101

United StatesNaval Postgraduate School

THESISFREQUENCY RESPONSE ANALYSIS AND DESIGN

OFSINGLE-VALUED NONLINEAR SYSTEMS

USING THE PARAMETER PLANE

by

John Philip Davis

June 1969

Tlni& document kcu been approved {^on. public >ie-

lzcu>£ and tale.; itt> dUVUJoutLon -U wlunltdd.

v rREY ,CALIF. 9394U

Frequency Response Analysis and Designof

Single-Valued Nonlinear SystemsUsing the Parameter Plane

by

John Philip DavisLieutenant

(junior grade), United States Navy

B.S., United States Naval Academy, 1968

Submitted in partial fulfillment of therequirements for the degree of

MASTER OF SCIENCE IN ELECTRICAL ENGINEERING

from the

NAVAL POSTGRADUATE SCHOOLJune, 1969

ABSTRACT

Frequency response techniques are a valuable tool in

the analysis and synthesis of linear systems. Extension

of these techniques is made to analyze and design systems

with a single-valued nonlinear element. The relationship

between the characteristics of a nonlinear element and the

frequency of the system is developed by simple calculations

and a digital computer program.

LIBRARY*L POSTGRADUATE SCHOOLREY, CALIF. 93940

DUDLEY KNOX LIBRARYNAVAL POSTGRADUATE SCHOOLMONTEREY, CA 93943-5101

TABLE OF CONTENTS

I

.

INTRODUCTION

II. INTRODUCTION TO NONLINEAR FREQUENCY RESPONSE

ANALYSIS USING THE PARAMETER PLANE

A. PREFACE

B. ANALYSIS OF FIRST EXAMPLE

C. STEPS IN THE ANALYSIS

D. FURTHER INVESTIGATION OF THE JUMP

RESONANCE EFFECT

III. FURTHER EXAMPLES OF NONLINEAR FREQUENCY

RESPONSE ANALYSIS

A

.

INTRODUCTION

B. FIRST EXAMPLE

C. SECOND EXAMPLE

D. THIRD EXAMPLE

E. FOURTH EXAMPLE

F. FIFTH EXAMPLE

G. SIXTH EXAMPLE

IV. THE EFFECT OF CHANGING THE INPUT TO THE

SYSTEM

A. INTRODUCTION

B. THE EFFECT OF CHANGING THE INPUT WITH

A SATURATION ELEMENT IN THE SYSTEM

C. THE EFFECT OF CHANGING THE INPUT WITH

A DEAD ZONE ELEMENT IN THE SYSTEM

13

17

17

17

39

41

50

50

50

58

63

74

82

90

101

101

101

111

V. DESIGN TECHNIQUES 120

A. FIRST DESIGN PROBLEM 120

B. SECOND DESIGN PROBLEM 140

VI. DISCUSSION, CONCLUSIONS, AND AREAS OF 147

FUTURE STUDY

A. DISCUSSION 147

B. CONCLUSIONS 147

C. FUTURE AREAS OF STUDY 149

APPENDIX A. ANALOG SIMULATION 151

COMPUTER PROGRAM-PARAMS 155

LIST OF REFERENCES 175

INITIAL DISTRIBUTION LIST 176

FORM DD 1473 179

4

LIST OF TABLES

Table Page

©II. la ol and £ as Functions of -g— for the 27

inSaturation Element

©II. lb ol and j3 as Functions of -g for the 27

inDead Zone Element

©II. 2 Of as a Function of «s— for the 46


©III.l ol as a Function of -ar— for the 53

inSaturation Element in Example #1

©III. 2 j3 as a Function of @-— for the Dead 62

inZone Element in Example #2

©111.

3

a as a Function of -s— for the Dead 70in

Zone Element in Example #3©

111.

4

ft as a Function of -^— for the 78in

Saturation Element in Example #4©

III. 5 a as a Function of ar— for the Relay^in

Element in Example #5

Dead Zone Element

inin

86

©III. 6 ol as a Function of q—— for the Coulomb 94

inFriction Plus Stiction in Example #6

©IV. 1 O as a Function of Tap— and ©. for the 108


©IV. 2 j8 as a Function of 75- and ®. for the 116

©V.l £ as a Function of 7s— for the Relay 135

Hinwith Dead Zone

LIST OF ILLUSTRATIONS

Figure Page

2.1 Position Servomechanism of Chapter 2 18

2.2 Block Diagram of the Servomechanism of 20

Chapter 2

2.3a Saturation Element of Example in Chapter 2 22

2.3b Dead Zone Element in Chapter 2 23©

2.4 0i versus -g— Magnitude for a Nonlinearity 30in

in the Forward Path

2.5 Closed Loop Frequency Response of the 32

Servomechanism with a Nonlinearity in the

Forward Path

2.6 Open Loop Frequency Response for 35

Saturation and Dead Zone Elements in the

Forward Path©

2.7 £ versus -^ Magnitude for a Nonlinearity 36in

in the Feedback Path

2.8 Closed Loop Frequency Response of the 38

Servomechanism with a Nonlinearity in the

Feedback Path

2.9 Block Diagram of a Servo with Saturation 42©

2.10 0i versus -q- Magnitude for a Servo with 44in

Saturation

2.11 Closed Loop Magnitude Frequency Response 47

for a Servo with Saturation

Figure Page

2.12 Closed Loop Phase Frequency Response 49

for a Servo with Saturation

3.1 Block Diagram of Example #1 where ol 51

Represents the Variable Gain of the

Saturation Element

3.2 Saturation Element of Example #1 53

3.3 OL versus -g— Magnitude for Example #1 56in

3.4 Closed Loop Frequency Response of 57

Example #1

3.5 Block Diagram of Example #2 where j8 59


Dead Zone Element

3.6 Dead Zone Element of Example #2 60©

3.7 £ versus -g— Magnitude for Example #2 64in


Example #2

3.9 Block Diagram of Example #3 where OL 66


Dead Zone Element

3.10 Dead Zone Element of Example #3 68

3.11 OL versus -@-— Magnitude for Example #3 71in


Example #3

3.13 Block Diagram of Example #4 where £ 75


Saturation Element

Figure Page

3.14 Saturation Element of Example #4 76

3.15 j3 versus -§— Magnitude for Example #4 80in


Example #4

3.17 Block Diagram of Example #5 where & 83

Represents the Variable Gain of the Relay

3.18 Ideal Relay of Example #5 84

3.19 Oi versus -js— for Example #5 88in

3.20 Open Loop Frequency Response of Example #5 89

3.21 Block Diagram of Example #6 where 0L 91

Represents the Variable Gain of Coulomb

Friction Plus Stiction Element

3.22 Coulomb Friction Plus Stiction Element of 92

Example #6©

3.23 Oi versus -g Magnitude for Example #6 95in

3.24 Open Loop Magnitude Frequency Response 97

of Example #6

3.25 Open Loop Phase Frequency Response of 98

Example #6

3.26 Closed Loop Magnitude Frequency Response 99

of Example #6

3.27 Closed Loop Phase Frequency Response of 100

Example #6

4.1 Block Diagram of Servomechanism in 102

Chapter 2

Figure Page©

4.2 Cl versus -gz— with Constant cc Curves and 104in

Constant ©. Curvesin

4.3 Saturation Element in Servomechanism 105

4.4 Closed Loop Frequency Response for 110

Various ®.in

4.5 Block Diagram of Servomechanism with 112

Dead Zone in the Feedback Path

4.6 Dead Zone Element of Second Example 113©

4.7 j3 versus q- Magnitude for Second Example 118in

4.8 Closed Loop Frequency Response for Various 119

©. in a System with Dead Zonein •*

5.1 Type 1 Third Order Feedback System 121

5.2 Third Order Feedback System with 12 3

Tachometer Feedback©

5.3 K, versus tr Magnitude for Design Problem 124in

5.4 Third Order System with Linear/Nonlinear 126

Compensator in the Tachometer Feedback Loop©

5.5 versus 75 Magnitude for Design Problem 127in ©

5.6 versus 75 Magnitude for the Design 130in

Problem

5.7 Plot of © -Signal out of the Element, versus 132El

© -Signal into the Element, with Restricted

Design Zone and Proposed Designs©

5.8 j3 versus -g— Magnitude with Proposed Relay 136in

with Dead Zone Constant ®. Curvein

10

Figure Page

5.9 Closed Loop Frequency Response of Problem 137

with Proposed Design Solutions

5.10 Block Diagram of Second Design Problem 142

5.11 Desired Closed Loop Frequency Response 143

of Second Design Probleme

5.12 0C versus -g— for Second Design Problem 144in

5.13 Characteristics of the Nonlinear Element 146

of Second Design Problem

A.l EAI TR-20 Electronic Comparator 153

11

ACKNOWLEDGEMENTS

The author wishes to express his sincere appreciation

and indebtedness to Doctor George J. Thaler for the topic,

guidance and assistance which he provided during the pur-

suit of this study and the preparation of this thesis. The

author would also like to express his extreme indebtedness

to his wife, Nancy Anne, for her invaluable help in finish-

ing this thesis.

12

I. INTRODUCTION

Frequency response techniques are a valuable tool in

the analysis and synthesis of linear systems. Reasons for

this are, in part, the ease of presentation and manipula-

tion of block diagrams and transfer functions, and in part,

the ease of computation and interpretation of Bode diagrams

and Nichol charts. It is therefore desirable to extend

these techniques to include nonlinear systems. Block dia-

grams with nonlinear elements can be handled with the

addition of some new rules. Handling transfer functions

that include nonlinearities presents more of a problem.

The nonlinear element's characteristics must be described

in terms of frequency so that they can be treated mathemat-

ically with the linear part of the system. This thesis

develops a technique which shows the relationship between

the nonlinearity and the frequency response of the system.

It then uses this relationship to perform analysis and

synthesis of systems that contain nonlinearities.

Several assumptions are made and some conditions must

be satisfied. The most important condition is that the

nonlinearity be a single-valued nonlinearity. A single-

valued nonlinearity is one that for any given input has

only one output. This condition eliminates elements such

as backlash, negative deficiency, two-position relay, relay

with hysteresis, etc. By making this condition, a

13

single-valued nonlinearity can be expressed as a variable

gain device, with the gain a function of the signal into

the device.

Two other main conditions are required. The first is

the condition that the input to the system be a pure sine

wave of constant amplitude,. The second condition is that

the input signal to the nonlinear component is a pure sine

wave.

In order to describe the nonlinearity as a function of

frequency, it must first be described as a variable gain

device. This gain is a function of the signal amplitude

into the element. Using the PARAMS program, a plot of the

gain versus the signal amplitude can be made. Each curve

in the resulting family is for a fixed frequency. With

this plot the gain can be described as a function of fre-

quency. Using this relationship between gain and frequency,

the analysis can be completed on a frequency response plot

with constant gain curves.

The analysis and synthesis techniques use the PARAMS

program (reproduced in the computer print-out section) to

make two types of plots. The first type of plot is a plot

of nonlinear gain versus the signal amplitude into the non-

linear element with constant frequency. The second type of

plot is a plot of the magnitude (or phase) of the open or

closed loop function versus frequency. Each curve in the

family is for a fixed value of gain for the nonlinear

element.

14

A brief description of the operation of the PARAMS pro-

gram is presented below. Let a transfer function be repre-

sented by,

*<-)-!$• (i.D

Where N(s) is of the form A +A,s+A„s + A .. sn_1

+ A sn

o 1 2 n-1 n

and D(s) is of the form B +B, s+B s +«««B , s +B s .v ' o 1 2 m-1 m

Each A, and B, can be represented by,

Ak

= ck

+ L^a + Ek £+ F

kaj3 (1.2a)

E^ = Gk+ E^Ci + ik

+ Jk«0 (1.2b)

Where OL and j9 are two variable parameters of the

system. T(s) can be represented by,

N (s) + N (s)

T < S> = D (s) + D (a)

(1 * 3a)

e o

and T(-s) by,

N (s) - N (s)

T '-S» - D

e(s) - D°(s)

(1 - 3b)

e o

Where N (s) is the even order terms of the numeratore x

'

of T(s), N (s) is the odd order terms of the numerator of

T(s), D (s) is the even order terms of the denominator of

T(s), and D (s) is the odd order terms of the denominator

of T(s)

.

15

The mangitude squared function is,

[N (s)r - [N (s)]M = Ct(s)] Ct(-s)] = — 5

[D (s)]^ - CD (s)](1.4)

Each term of equation (1.4) is a even polynomial in s

Letting s = jw, equation (1.4) can be reduced to,

N

K^k

M 2 k=0M =m (1.5)

Iv*k=0

Where A, and B, are given by equations (1.2a) and

(1.2b)

.

By letting Oi or j3 be a constant, plots of the remain-

ing variable parameter (j8 or Oi) versus magnitude can be

made with frequency constant. In like manner, plots of

magnitude versus frequency can be made with a constant

parameter value.

Phase calculations can be made by utilizing the for-

mula,

= tan-1 N (s)D (s) - N (s)D (s)

o v ' e e oN (s)D (s) - N (s)D (s)Lee oo_i (1.6)

In this thesis Oi or )3 represents the variable nonlinear

gain.

Glavis, G.O., Frequency Response in the ParameterPlane , Master's Thesis, Naval Postgraduate School, 1967.

16

II. INTRODUCTION TO NONLINEAR FREQUENCY RESPONSE

ANALYSIS USING THE PARAMETER PLANE

A. PREFACE

As an introduction to the technique of nonlinear fre-

quency response analysis, a fourth order system will be con-

sidered. The analysis is divided into two parts. One part

is the analysis of the system with a nonlinear element in

the forward path. The second part is the analysis of the

system with a nonlinear element in the major feedback path.

Each part will be analyzed for a saturation element and

then analyzed for a dead zone element. All the assumptions

made in the first chapter are valid for this example and

all other examples used.

The second section of this chapter looks closer at the

jump resonance effect that a nonlinear element can cause.

The example taken is a simple second order servo with a

saturation element in the forward path. The analysis tech-

niques are the same in this example as were those in the

first example of this chapter.

B. ANALYSIS OF THE FIRST EXAMPLE

The first example of this analysis is a fourth order

system with a nonlinear element in either the forward path

or the main feedback path. The fourth order system chosen

for this example is a simple position servomechanism which

is shown in Figure (2.1).

17

<±>

QPiS

W Eh55 55M Ixl

§ a55 M

I I

-vvyvvv

VWW-4

s$

/'"V

Figure 2.1 POSITION SERVOMBCHANISM EXAMPLE OF CHAPTER 2

18

The transfer function of the amplidyne is,

kq

G (s) = j-3 j— (2.1)(s + i-) (S + ^-)

f q

Xf

Xqwhere r - — and T - —f H q

The motor-load combination transfer function is,

kG_(s) = 2L

1— (2.2)

ILL / X v

s(s + —

)

1

Where T, is the motor-load time constant and is pre-

determined.

The error detector has a gain of k .

e

Putting the position servomechanism into block dia-

gram form is done and is shown in Figure (2.2). The non-

linear element in the forward path is fed by the error

signal © . The output signal, © feeds the nonlinear ele-6 3.

ment in the main feedback path. These nonlinear elements

are represented by blocks with their associated variable

gain. Qt is the variable gain for the nonlinear element in

the forward path and fi is the variable gain for the non-

linear element in the main feedback path. The two nonline-

ar elements are not in the system at the same time.

Analysis will be done for the nonlinearity in the forward

path and then for the nonlinearity in the main feedback

path.

19

JJ

3

COoH

CM00VOo\

o +oSO CO

CDrH+

QCL

to

®°

a•"

^vQJ©

r\¥ 'i

Figure 2.2 BLOCK DIAGRAM OF THE POSITION SERVOMECHANISM

OF CHAPTER 2

20

Assume that — = 10.0, — = 5, ^- = 3, and k k k =600.Tf

Tg

T1 q m e

The characteristic equation now becomes,

s4

+ 18s3

+ 95s2

+ 150s + 600ttj3 = 0.0 (2.3)

And the closed loop transfer function is,

©out 600a

,..^ = ~4 3

2 1(2 * 4)

in s + 18s + 95s + 150s + 600tt/3

When analyzing the system with a nonlinearity in the

forward path, j8 will be equal to unity. The converse is

true when analyzing with the nonlinearity in the feedback

path. OL will then be equal to unity.

Many types of single-valued nonlinearities could be

considered here. Two types are used in this example and in

examples that will follow mainly for simplicity sake. These

two types are saturation and dead zone. For this example

the saturation element limits at 10.0. as shown in Figure

(2.3a). Figure (2.3b) shows the dead zone element where

the dead zone ends at 10.0. The assumed magnitude of ©.J in

is 20.0. It could be any magnitude; however it must be

specified for the analysis because the frequency response

is a function of the magnitude of ©. . Because the gain of

the nonlinear element is a function of the signal going into

it, what is desired is a relationship between the gain of

the nonlinear element and the input signal. However, it is

hard to describe the signal going into the nonlinear ele-

ment as a function of the nonlinear gain. It is easier to

21

14-1

OJJ

GCD

ECD

i-H

CD

G

•h ,Gen -u

CD

JJ

O

c•H

JJ-I c

£ I•H nHCO QJ

Figure 2.3a Saturation element of example in chapter 2

22

iw

4J•U CP CD

o eCD

iH iH03 CD

C otn CD •

•h ,n oCO 4-> —1—

ot

or-H

,1

Figure 2.3b Dead zone element in chapter 2

23

develop a relationship between the gain of the nonlinear

nlelement and -s— , where © , is the signal going into thein

nonlinear element and ©. is the input into the system.in J

For this example, © , = © for the case of the nonlineari-nl e

ty in the forward path, and © , = © for the nonlineari-* r nl out

ty in the main feedback path. This may be seen in figure

(2.2) .

For the case of the forward path nonlinearity, the

gain, OL, may be described as,

a* = ©^ (2.5)

e

Where ©, is the signal out of the nonlinear element.

Dividing both the numerator and the denominator of the

right-hand side of equation (2.5) by ©. ,

_. d in /0 a y.

01 = ©/© (2 ' 6)e/ in

Referring to figure (2.3) for the saturation element,

©, = 10 when © is greater than 10. When © < 10, ©, = ©b e 3 e b e

and a is equal to unity.

Substituting ©. = 20 into equation (2.6) yields,

0/20

e in

24

Two cases now exist. When ^ 10, © = 0, ande e b

© /20a = WT^ = 1-0 ® * 10 (2.8a)© /20

When © > 10, ©, = 10 ande b

e

<* = Q-y!; ©e

> 10 (2.8b)e/ in

From equations (2.8a) and (2.8b), a table may be drawn

up giving the gain of the saturation element as a function©

of -=— . This table would be the same for the saturation©•in

element in the main feedback path, by replacing © by © ,

©by © and Oi by )9. Equations (2.8a) and (2.8b) become,

© ,.720our© _/©.

x ' u"out"our in

= A°Ut

/ft = 1.0 ® < 10.0 (2.9a)

'=

© 'ye. out* 10 '° < 2 ' 9b >

out in

This table for the saturation element is table (II. la)© ©

To use it, enter the left-hand column with q— or -g andin in

read the corresponding value of Oi or j8 respectively.

In like manner the dead zone device can be dealt with.

Referring once again to figure (2.2)

a = ©^ (2.5)e

25

Dividing again by 0. and substituting,

V in V20a =

© /©= © -m (2.6), (2.7)

e in e/ in

Looking at figure (2.3b), it is seen that for all input

signals into the dead zone element, in this case © , less

than 10.0, the output, ©, , is zero. Therefore,

Qi = q %20> = 0.0 ©e

^ 10.0 (2.10a)e/ in

And for © greater than 10.0,e 3

© /20<* = ©/© e

* 10 -° (2.10b)e in

For the dead zone in the main feedback path, substitu-

tion once again obtains,

* - ©0//© = 0.0 ©out * 10 '° (2 ' lla)

out in

© /20.

* r-7*r out> 10 -° (2 - llb)

our in

26

From equations (2.11a) and (2.11b) and figure (2.3b),

e out Saturation Dead ZoneElement Element

<S)01

<=>

in in 0L or j3 0L or $

0.00 1.000 0.0000.25 1.000 0.0000.50 1.000 0.0000.75 0.670 0.3331.00 0.500 0.500l e 25 0.400 0.6001.50 0.333 0.6661.75 0.28 0.7152.00 0.25 0.7502.25 0.22 0.7782.50 0.20 0.8002.75 0.18 0.8173.00 0.17 0.8343.25 0.15 0.8463.50 0.14 0.8573.75 0.13 0.8684.00 0.125 0.8754.25 0.118 0.8804.50 0.112 0.8874.75 0.105 0.8945.00 0.100 0.9005.25 0.095 0.9055.50 0.091 0.9105.75 0.087 0.9146.00 0.083 0.9176.25 0.080 0.9206.50 0.077 0.923675 0.074 0.9267.00 0.071 0.9297.50 0.067 0.9328.00 0.063 0.9388.50 0.059 0.9419.00 0.055 0.9459.50 0.053 0.948

10.00 0.050 0.950

©ei one: o-F -x

inSaturation Element b.) OL and j3 as Func-

tions of gPu

for the Dead Zone Elementin

27

a table similar to table (II. la) can be drawn up for the

dead zone element. This table is table (II. lb). Once©

again enter with either -*— or -g and QL or j3 can bein in

read off respectively.

The next step in obtaining the frequency response is

to find a relationship between the nonlinear gain, Ot or $,

and the frequency. A relationship between the nonlinear

gain and the signal into the element was just shown. Using

the PARAMS program, a relationship between OL or |8 and fre-

quency can be found. The PARAM-5 part of the program does

this. Given a transfer function, it will draw a plot of

gain, OL or jS, versus the magnitude of the transfer function

with constant frequency lines drawn.

The case of the nonlinearity in the forward path will

be considered first. In this case j3 = 1.0 and © = © ..a out

Looking at figure (2.2),

0=0. - © (2.12)e in out

Divide both sides of equation (2.12) by ©. ,

© ©Tg- = 1 - ^ (2.13)in in

©gPu has already been computed as equation (2.4)

©out eoooi , „.© = ~ 5 n " U-4)in ;

4+ 18s

3+ 95s

2+ 150s + 600aj8

28

Substituting equation (2.4) into equation (2.13) and

letting j3 = 1.0.

© 4 3 2e s^ + 18s

J+ 95s + 150s

73) a o ? I*. 14 J

in s + 18s + 95s + 150s + 6008

Using equation (2.14) and letting OL vary between 0.0

and 1.5, a plot as described in the previous paragraph is

obtained by the PARAM-5 subprogram of the PARAMS program.

This plot is figure (2.4), with OL, the nonlinear gain

in the forward path as the ordinate and 75— as the abscissa,in

Constant u> lines, W = 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6,

1.8, 2.0, 2.2, 2.4, 2.6, 2.8, 3.0, 3.2 and 3.4, were drawn

by the PARAMS program. Using table (II. la) and table

(II. lb), constant ©. lines are drawn on figure (2.4) for

the saturation element and for the dead zone element. The©

constant ©. lines are drawn using OL and 75— as the coor-in

dinates.

After drawing the constant ©. curves on figure (2.4),

it is seen that these curves intersect the constant omega

curves in several places. The intersection of the constant

©. curves with the constant omega curves, with the asso-

ciated value of 0L at each intersection, determines a rela-

tionship between 0L, the nonlinear gain, and 03, the fre-

quency of the overall system. With this relationship

between 0L and (jO, the total frequency response will be ob-

tained. Figure (2.4) shows a pair of relationships, one

29

a

0.0

© inFigure 2.4 OL versus -g— magnitude for a nonlinearity in the

forward path in

30

for the saturation element and the other for the dead zone

element. These two are in no way related to each other.

They are both shown on the same plot so as to analyze two

different cases at one time.

The final step in the analysis is obtaining the fre-

quency response itself. Once again the PARAMS program is

used. The PARAM-7 subprogram of the PARAMS program gives

a frequency response plot with constant Q or 3 lines drawn

on it. Because (for this analysis) the closed loop fre-

®outquency response is desired, a plot of -j* magnitude (dB)in

versus frequency with constant OL curves is obtained. This

is figure (2.5) with OL curves of OL = 0.0, 0.1, 0.2, 0.3,

0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, and

1.5 drawn on it.

Now using the relation that was found in figure (2.4)

between 0L and CO through the intersection of the constant to

curves and the constant to curves, these points are plotted

on figure (2.5). The coordinates to transfer these points

from figure (2.4) to figure (2.5) are 0L and to. Where there

is no exact OL or w line on figure (2.5), interpolation is

used.

When looking at figure (2.5) it should be noted that

the saturation frequency response shows some jump resonance

around its resonance peak. This is characteristic of sa-

turation elements and can be predicted from figure (2.4).

31

VT\ O

•§|©

V>>

I

Figure 2.5 Closed loop frequency response of the servo-

mechanism with a nonlinearity in the forward path

32

To do this, trace the constant ©. saturation curve in

figure (2.4) from left to right. Note that it intersects

the constant iC = 1.4 curve twice giving two values of non-

linear gain, &, for Ui = 1.4. This is part of the jump re-

sonance effect and will be looked at more closely in Chap-

ter 4.

To test the results of this graphical analysis, an

analog simulation of the system was performed and the

actual frequency response was obtained. How this simula-

tion was performed and how the nonlinear elements were

simulated is explained in Appendix A. The analog results

are plotted on figure (2.5) as 'x'. The analog simulation

was close enough to justify the conclusion that the graph-

ical analysis was accurate in predicting the frequency

response. Any difference between the predicted response

and actual response is due to errors in graphical construc-

tion, reading of the graph and tolerance in the simulating

equipment.

The main effort of this thesis is directed to closed

loop frequency response. However, the open loop frequency

response may be obtained in the same manner as the closed

loop response. The means of obtaining the open loop res-

ponse is in the final step of the analysis. When using

the PARAM-7 subroutine of the PARAMS program, instead of

using the closed loop transfer function, the open loop

transfer function should be used. The plot that should be

33

obtained is the open loop magnitude (dB) versus frequency

with constant OL curves drawn on the plot by the program.

For this first case the plot is figure (2.6). The phase

curve is superimposed for more information. The next step

is the same as in the first case. Using the coordinate

values of OL and 00 found in figure (2.4), redraw the con-

stant ©. curves on figure (2.6). The result is the open

loop frequency response for a system with either saturation

or dead zone in the forward path.

The next case in this example is a nonlinear element

in the main feedback path. In this case QL = 1.0 and ©r out

is the signal into the nonlinear element as shown in figure

(2.2). Equations (2.9a), (2.9b), (2.11a), and (2.11b) have

been developed for the nonlinear element in the main feed-

back path and the results tabulated in table (2.1a) for a

saturation element and table (2.1b) for a dead zone element.

Using the PARAM-5 subprogram of the PARAMS program, a

plot of j3, the nonlinear gain in the main feedback path,

®outversus the magnitude of ** is obtained in the same mannerin

as for figure (2.4). This plot is figure (2.7) and shows j8

®outbetween 0.0 and 1.5 as the ordinate and -^ as the abscissain

with constant 05 curves of O) = 0.4, 0.6, 0.8, 1.0, 1.2, 1.4,

1.6, 1.8, 2.0, 2.2, 2.4, 2.6, 3.0, 3.2, and 3.4 drawn by

the program on the plot.

34

oOnI

o oVN. oc^ 00 lr«

Figure 2.6 Open loop frequency response for a saturationand dead zone elements in the forward path

35

1.4

1.2

1.0

0.0

outr~in

inFigure 2.7 $ versus -r— magnitude for a nonlinearity in

the feedback path

36

The next step in the analysis is the same as in the

first case. Using the values in table (Il-la) and table

(Il-lb) , constant ©. curves for saturation and dead zone

are drawn on figure (2.7). The coordinates for drawing

®outthese two curves on figure (2.6) are j9 and -g . Thein

intersection in figure (2.6) of the constant ©. curves3 v in

with the constant 00 curves with an associated for each

intersection, gives the relationship between the nonlinear

gain, j3 and 00 for the system.

The final step in this analysis of the second case is

the obtaining of the closed loop frequency response. As in

the first case, the PARAM-7 subprogram of the PARAMS program

is used. Using equation (2.4) with a. = 1.0 figure (2.8) is

®outobtained with the magnitude of -pj (dB) as the ordinate andin

frequency as the abscissa. Constant j3 lines of # = 0.0,

0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2,

1.3, 1.4, 1.5 are drawn on the plot by the program.

Using the coordinate of the intersection on the con-

stant ©. curves and the constant 00 curves, that is j8 and oo,in

the constant ©. curves are drawn on figure (2.8). Thesein 3

curves on figure (2.8) give the closed loop frequency re-

sponse of the overall system for a saturation element in

the main feedback path and for a dead zone element. Note

that there is no jump resonance in either response. This

is as predicted by looking at figure (2.7) and observing

that neither constant ©. curves intersects a constant omegain

37

Figure 2.8 Closed loop frequency response for servo-mechanism with a nonlinearity in the feedback path

38

curve more than once. This could have occurred in both

cases, saturation and dead zone, if the constant 0. curvesin

had been shifted sufficiently to the right in figure (2.7).

The shifting to the right of the constant ©. saturation3 3 in

curve corresponds to raising the value at which the signal

saturates. The shifting of the constant ©. dead zone3 in

curve to the right corresponds to increasing the area of

dead zone.

The results in this second case, a nonlinearity in the

main feedback path, were also checked by analog simulation.

The simulation results are plotted with 'x' and correspond

enough to verify that the graphical analysis is correct.

C. STEPS IN THE ANALYSIS

The steps in the analysis are listed below for ease in

following the process. The use of the PARAMS program makes

this analysis realizable because of its speed and ease of

calculation. At the present time, the PARAMS program is

limited to a 49th order system.

1. Draw the block diagram of the system, with the

nonlinear element as a separate block. In order to use

this technique, the nonlinearity must be a single-valued

type.

2. Write the transfer function of the closed loop

system. Also write the transfer function of the signal pre-

ceding the nonlinearity with respect to the input signal

39

into the system. In some cases these two transfer func-

tions may be the same as in the case in this chapter where

the nonlinearity was in the main feedback path.

3. Using the PARAMS program, or its equivalent, ob-

tain a plot of the nonlinear gain versus the magnitude

transfer function of the signal into the nonlinearity.

This plot should have drawn on it constant to curves for

the frequencies of interest.

4. Obtain a relationship between the nonlinear gain

and magnitude of the transfer function of the signal into

the nonlinearity. This can be done several ways, one of

which is described in this chapter.

5. Plot this relationship found in step 4 onto the

plot drawn in step 3 using the nonlinear gain and transfer

function magnitude as the coordinates. This is the constant

©. curve,in

6. The intersection of the constant ©. curve and thein

constant to curves gives an associated value of nonlinear

gain for each intersection. These intersections give the

relationship between the nonlinear gain and frequency. In

some cases there may be more than one associated value of

gain for a certain frequency. This can be ascribed to a

jump resonance effect.

7. Again using the PARAMS program or its equivalent,

obtain a standard frequency response plot, either open or

closed loop, with constant nonlinear gain curves drawn on

40

it. The values of nonlinear gain curves must correspond to

the range of nonlinear gain found in step 6.

8. Using the values of nonlinear gain and frequency-

found in step 6 as coordinates, replot the constant ®.

curve on the plot found in step 7. This gives the closed

or open loop frequency response - whatever the case may

be - of a system with a nonlinear element to a constant

magnitude input signal.

D. FURTHER INVESTIGATION OF THE JUMP RESONANCE EFFECT

The frequency response for the system with saturation

in the forward path, figure (2.5), shows a jump resonance

around CO = 1.4. This jump resonance is to be expected with

a nonlinear device. In Chapter 4 this effect will be shown

for a dead zone element. To illustrate this jump resonance

more clearly, another example will be shown here. The ex-

ample is a second order feedback system which is used as an

2example in a nonlinear control textbook. Its block diagram

is figure (2.9)

.

By letting Oi represent the variable gain of the satura-

tion element, the closed loop transfer function may be

written as,

Oout 200a (2.15)W~~ ~ ~2in s + s + 200a

2

Thaler, G. J. and Pastel, M.P., Analysis and Designof Nonlinear Feedback Control Systems , p. 198, McGraw-Hill,1952.

41

XJ

.0

OoCM

+03

«°

(N

£*_

.O .0

Figure 2.9 Block diagram of a servo with saturation

42

What is desired is the frequency response with a con-

stant input magnitude. 0. is chosen to be ©. =2.0.in in

Following the steps in the analysis technique, the

transfer function of the signal preceding the nonlinear

element with respect to ©. , that is -*—, is derived

in

© = 0. - © (2.16)e in out v'

Dividing by ©. ,

© ©e _ t

out , __.-©— - 1 - -@ (2.17)in in

And substituting equation (2.15) into equation (2.17),

© 2

#" =2

S S(2.18)

in s + s + 2000

By using the PARAMS-5 subprogram of the PARAMS and©

equation (2.18), a plot of Oi versus -s— with constant u>

incurves is obtained. This plot is figure (2.10).

©eNext the relation between q— and Oi is found.in

Referring to figure (2.9)

©e

Dividing both the numerator and denominator of the «

right-hand side of equation (2.19) by 0. t

a/e.a = © /©

in-

( 2 - 2 °)e7 in

43

!.#

1.2

1.0

OL

0.0

6 inFigure 2.10 Of versus m— magnitude for servo withsaturation in

44

and when © < 20, a = 1 and when © )| ZO, «=^.0 as thee fe Bio

element goes into saturation. Two cases exist for ©. =2.0.in

©a = 1.0 -^- z 1.0 (2.21a)

in

©a = & 75 #~ * 1 -° (2.21b)

e' in in

With equations (2.21a) and (2.21b), a table is con-

es true ted giving a for the corresponding values of q— .

© inThis table is table (II. 2) where 0.0 £ -zp- £ 10.0.

inUsing the values in the table, the constant ©. curve^ in

is plotted on figure (2.10). The intersection of this con-

stant ©. curve and the CO curves in figure (2.10) givesin r» \ / 3

the relation between CO and Qi for a constant ©. =2.0.in

The closed loop frequency response of the servo can

now be obtained for a ©. =2.0. Using the PARAM-7 sub-in

®outprogram and equation (2.H), a plot of -g (dB) versus CO

inis obtained. This is figure (2.11) and has constant 0L

curves of a = 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4,

0.45, 0.5, 0.55, 0.6, 0.7, 0.8, 0.9 and 1.0 plotted on it.

Using the coordinates of 0i and 03 for the constant ®in

curve

in figure (2.10), the constant ©. curve is replotted ontoin ^

figure (2.11) . As may be seen, there is a large jump re-

sonance effect between co = 6.0 and CO = 10.0.

The jump resonance effect can also be seen in the

phase curve of this example. The technique is similar to

that of drawing the magnitude frequency response. Using

45

e

0.ina

0.0 1.0

0.5 1.0

1.0 1.0

1.5 0.667

2.0 0.500

2.5 0.400

3.0 0.333

3.5 0.286

4.0 0.250

4.5 0.222

5.0 0.200

5.5 0.182

6.0 0.167

6.5 0.154

7.0 0.143

7.5 0.133

8.0 0.125

8.5 0.118

9.0 0.111

9.5 0.105

10.0 0.100

©Table II. 2 & as a Function of -zr— for the©in

Saturation Element

46

Figure 2.11 Closed loop magnitude frequency response for a

a servo with saturation

47

the PARAM-7 subprogram again, a plot of the phase of J3

inversus omega with the same constant & curves is drawn.

This is figure (2.12). Using the coordinates of ol and Ui

from figure (2.10), the constant ©. curve is drawn on the^ in

phase plot, figure (2.12). The jump resonance effect is

seen again, though not as pronounced.

48

Figure 2.12 Closed loop phase frequency response for theservo with saturation

49

III. FURTHER EXAMPLES OF NONLINEAR

FREQUENCY RESPONSE ANALYSIS

A. INTRODUCTION

In this chapter, six examples are worked in order to

further verify the validity of the analysis technique.

Saturation, dead zone, an ideal relay, and coulomb frction

plus stiction are the nonlinearities used in the examples.

These nonlinearities are placed in various positions in the

systems including the forward path and the minor feedback

path. The first four examples were checked by analog sim-

ulation. Also in all six examples ©. was assumed to be^ in

of magnitude ©. =10.0.3 in

B. FIRST EXAMPLE

The first example to be worked is a third order system

with a saturation element in the forward path. Its block

diagram is figure (3.1) where a represents the variable non-

linear gain of the saturation element. From figure (3.1),

Jut = 9«(3-1)

^in s + s + 16.25s + 90L

The saturation element has unity gain for © signals

less than 50.0. Signals into the element greater than 50.0

50

4J

inCM

•

t£rH

o +• CO

<T> +CM

en«—

*

w

.<D

Figure 3.1 Block diagram of example #1 where ft representsthe variable gain of the saturation element

51

are limited so that the output of the element is 50.0.

The input-output diagram of the element is figure (3.2'

Referring to figure (3 «, 1) „

" = ©- (3.2)e

Dividing both the numerator and the denominator of the

right-hand side of equation (3.2) by ©.

aye.

For a ©. =10.0, equation (3.3) can be divided into

two parts

a = 1.0 gp- ^ 5.0 (3.4a)in

©

e' in in

Equations (3.4a) and (3.4b) define the relationship©e • •

between d and -r— . It was seen in the previous chapter"in

that this relationship is necessary to obtain the frequen-

cy response. These equations are tabulated in table (II. 1) ,

©where the left-hand column is ;»— and a is the value of

©• «in ©e

nonlinear gain for the particular -~— .

in

52

Figure 3.2 Saturation element of example #1

53

Having obtained the relationship between ot and -ap— ,

inthe relationship between ot and lc is found. Referring to

figure (3.1),

© = © _ ©e in out

© © - © ©e in ou t , out ,_ ...w = —©:

= 1 ~ a

—

(3 - 5)in in in

Substituting equation (3.1) into equation (3.5),

© 3 2e s + s + 16.25 . .

© 3 2 ' • '

in s + s +16.25 + 90?

Using equation (3.6) and the PARAM-5 subprogram,

©figure (3.3) is obtained where Ot is the abscissa and e

©.

is the ordinate. Constant to curves are drawn on figure

(3.3) .

A constant ©. curve is then drawn on figure (3.3)in qusing the coordinates of ot and -g— taken from table (III.l)

inThe intersection of the constant ©. curve and the to

in

curves gives the relationship between Ot and o> for this

analysis

.

The next step is drawing the closed loop frequency-

response. A plot is made, using equation (3.1) and the©

PARAM-7 subprogram, of -^ (dB) versus to. This plot isin

figure (3.4) and has constant Ot curves on it over the

range of Ot that is used. Using the coordinates of Ot and Ui

54

e

in

0.0 1.0

1.0 1.0

2.0 1.0

3.0 1.0

4.0 1.0

5.0 1.0

6.0 0.833

7.0 0.715

8.0 0.625

9.0 0.555

10.0 0.500

11.0 0.454

12.0 0.416

13.0 0.385

14.0 0.357

15.0 0.333

e

i

Saturation Element in Example #1

Table III.l a as a Function of -ap— for thein

55

1.4

1.2

1.0

a

0.0

Q inFigure 3.3 a versus -r- magnitude for example #1

in

12.0

56

Figure 3.4 Closed loop frequency response of example #1

57

from figure (3.3), the constant ©. curve is redrawn on^ in

figure (3.4). This constant ©. curve on figure (3.4) is

the closed loop frequency response of the first example

with the saturation element in the system. The analog sim-

ulation is denoted by 'x 1

.

C. SECOND EXAMPLE

The second example used for analysis is a third order

system with a dead zone element in the tachometer feedback

path. The block diagram of the system is figure (3.5)

where £ represents the variable nonlinear gain of the dead

zone element. From figure (3.5), the closed loop transfer

function is written,

©out _ 9 ,~ _.

73 ^ 2 * * '

in s + 16.25s + 9 + ]8s

The dead zone element is shown in figure (3.6). For

a © less than 4.0, ©,, the output, is zero. For a con-a d @

astant input of ©. = 10.0, this condition exists when W

inis less than 0.4. In equation form,

© ©* = w = w^ = °-° w~ * °- 4 (3 - 7)

a a in

From figure (3.6), it is seen that for © greater thana

4.0, ©, = 3 (© - 4.0). Thus,d a

© 3(© - 4.0)

= W = ^© ®a * 4 '° (3 ' 8)

a a

58

JJ

3^09

03

IT)

CN•

r-H

+(N ro

w V

00.

^ T)

c+ '

CT> en

®

N. 1

Figure 3.5 Block diagram of example #2 where j9 representsthe variable gain of the dead zone element

59

•12.0

--8.0

-- 4.0

4.0 8.0 ©

—4.0

--8.0

---12.0

Figure 3.6 Dead zone element of example #2

60

Dividing the numerator and denominator of the right-

hand side of equation (3.8) by ®. when ©. =10.0,^ 'in in '

3©

(3© - 120)/©. ©. " 12 *° ©a a in in a .

p- ©-7©- = —©~7© w~ °* 4

a / in a' in in

(3.9)

Equations (3.7) and (3.9) give the relation between j3

©and 75— . From these two equations, a table has been tabu-

in © ©lated of ? as a function 75— over the range of 75— desired,

in © inThis table is table (III. 2) where 75— is the left-hand

incolumn and ]8 is the value read out for the corresponding

©value of 75— .

in ©aHaving found the relationship between j3 and 75— , thein

relationship between £ and ixy is now found. Referring to

figure (3.5)

,

©a

out

*.out

= s (3.10)

(3.7)in s

3+ 16.25s + 9 + jSs

2

Multiplying equation (3.10) by equation (3.7),

© Q

W~ = "1 —2

(3 - 11}in s + 16.25s + 9 + j8s

61

©a

©.in

0.0 0.0

0.05 0.0

0.10 0.0

0.15 0.0

0.20 0.0

0.25 0.0

0.30 0.0

0.35 0.0

0.40 0.0

0.45 0.333

0.50 0.600

0.55 0.819

0.60 1.00

0.65 1.16

0.70 1.28

0.75 1.40

0.80 1.50

0.85 1.59

0.90 1.67

0.95 1.74

1.00 1.80

Table III. 2 j3 as a Function of 75— for the Deadin

Zone Element of Example #2

62

Using equation (3.11) and the PARAM-5 subprogram, a

plot of j8 versus the magnitude of -x— is obtained. This isin

figure (3.7) and has constant 40 curves of 03 = 1.0, 2.0,

3.0, 3.5, 3.7, 3.8, 3.9, 4.0, 4.1, 4.2, 4.3, 4.4, 5.0, 6.0,

7.0 and 10.0 drawn on it.

The constant ©. curve for the dead zone element ofin

this example has been drawn on figure (3.7). The coordi-

nates for this curve are £ and -q— , and were taken fromin

table (III. 2). The intersection of the constant ©. curvein

and the O) curves in figure (3.7) gives the relationship

between |3 and 03 for the system. Using this new relation-

ship, the closed loop frequency response can now be drawn.

Equation (3.7) and the PARAM-7 subroutine are used to6outobtain a plot of -g versus co with constant $ curves overin

the range of ]8 = 0.0, 0.1, 0.2, 0.4, 0.6, 0.8, 0.9, and

1.0. Using the coordinates of $ and CO for the constant ®.3 in

curve on figure (3.7), the constant ©. curve is redrawn

on figure (3.8). This constant ©. curve on figure (3.8)

gives the closed loop frequency response of this second

example with the dead zone element of figure (3.6) in the

tachometer feedback path. The frequency response was check-

ed by analog simulation and is denoted by 'x'.

D. THIRD EXAMPLE

The third example is a fourth order system with a dead

zone element in the forward path. The block diagram is

figure (3.9) where 0i represents the variable gain of the

63

$

9 inFigure 3.7 versus »— magnitude for example #2

in

64

i

I

rr

1 fc^ •

H\ ~*r •

-3Jr xn

t; #Q^ ^^

i

_x ">y ^pft)

1

1

<n

-Pnn

/^

IT

•

/ C^

n

>

c-HO

c 13

fC X

0}

I

•

;

C J-4 I\>J

f

f

1

i

L")n

*

a

I

00 • .is •" HI


65

4J

rH +n

r-H +en

©°

s

9°

rS:

h^ID

t

r-i

+to

CO

s^+ I'

Figure 3.9 Block diagram of example #3 where a representsthe variable gain of the dead zone element

66

dead zone element. is the signal going into the element

and is the signal coming out of the element. From

figure (3.9), the closed loop transfer function is,

out 14. 50L ._ 19 .

75 '432 32 \3'1-*)in s +13s +51s +63s+(s +10s +21s+14.5)tt

The dead zone element is shown in figure (3.10) where

the ordinate is ©, and the abscissa is . The dead spaced e

extends to © =50.0. For a © greater than 50.0, the gaine e

of the element is two. From figure (3.9),

= ^ (3.13)e

For © £ 50.0,e

e in

For © 2; 50.0, ©, can be written as,e b

©^ = 2(© - 50.0) = 2© - 100.0 (3.15)beeSubstituting equation (3.15) into equation (3.13),

2© - 100.0=

eq ©

es 50.0 (3.16)

e

67

-100.0 -50.04>

•100.0

--50.

%

50.0 100.0 ©

•• -50.0

---100.0

Figure 3.10 Dead zone element of example #3

68


hand side of equation (3.16) by ©. and realizing that whenin

© ^ 50.0, 7-5— = 10.0, equation (3.16) becomes,in

2©e

* =©7©,— #- * 5 -° (

3 - 17 )

#- - io -° ©

From equations (3.14) and (3.16) a table of a as a©

function of 75— is computed. This table is table (III. 3)© in

and has 75— in the left-hand column and the correspondingin

value of 0i in the right-hand column. The table gives the©

relationship between Gi and 75— for a constant ©. =10.0.™ in a,in ©After finding the relationship between 0i and 75— ,

inthe relationship between a and Cti is found. Referring to

©figure (3.9) the transfer function of 75— is,

in

®e 14.s 2+145s+304.5 . .

1ST '432 32 IJ.loJin s +13s +51s +63s+(s +10s +21s+14.5)tt

Using equation (3.18) and the PARAM-5 subprogram, a©

plot of 0L versus 75— is made. This plot is figure (3.11)in

and has LC curves of 05 = 0.0, 0.025, 0.05, 0.75, 0.1, 0.2,

0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 and 1.0 plotted on it.

The constant ©. curve is plotted on figure (3.11)in

using the coordinates of 0i and 75— tabulated in tablein

(III. 3). The intersection of this constant ©. curvein

69

eWin

a

0.0 0.0

1.0 0.0

2.0 0.0

3.0 0.0

4.0 0.0

5.0 0.0

6.0 0.333

7.0 0.57

8.0 0.75

9.0 0.89

10.0 1.00

11.0 1.09

12.0 1.17

13.0 1.23

14.0 1.28

15.0 1.33

16.0 1.37

17.0 1.41

18.0 1.44

19.0 1.47

20.0 1.50

Table III. 3 Ci as a Function of -g— for thein

Dead Zone Element of Example #3

70

a

© inFigure 3.11 a versus -r— magnitude for example #3

in

71

and the to curves, at the associated value of at, gives

the relationship between at and to for ©. =10.0.

The final step in the analysis is construction of the

closed loop frequency response. Using equation (3.12) in

®outthe PARAM-5 subprogram, a plot of -g (dB) versus CO, with

inconstant at lines, is made. This plot is figure (3.12).

The constant ©. curve is replotted onto figure (3.12) from

figure (3.11) using at and to as coordinates. This constant

©. curve in figure (3.12) is the closed loop frequency

response of example #3. An analog simulation of this ex-

ample was run and the results shown by 'x' on figure (3.12),

72

May/A'// / s

.4fW/7/./ " /

W/f/

/

•riVKV I+T~f


73

E. FOURTH EXAMPLE

The fourth example is a fourth order system with a

saturation element in a minor feedback path. The analysis

is for the closed loop response. Figure (3.13) shows the

block diagram of this example where j3 represents the vari-

able gain of the saturation element in the minor feedback

path. © is the signal going into the saturation elementa

and ©, is the signal coming out of the saturation element.

Referring to the block diagram of this example (figure (3 .13 ) ),

the closed loop transfer function is written,

©out 14.5 ,- , *

in s +13s +51s +63s+14.5+/3(s +10s +21s)

Where $ represents the variable gain of the saturation

element.

The saturation element for this example is figure (3.14).

© is the signal into the element and © is the signal out

of the element. The saturation element limits at © = 20.0.a

For a © greater than 20.0, the output, ©,, is equal to

20.0. From figure (3.13), may be written as,

(3.20)®d

a

For © less than 20.0, © = ©, and,a ad©

= -gp- = 1.0 ©a

* 20.0 (3.21)a

74

o

+

<X>

v

Figure 3.13 Block diagram of example #4 where fi representsthe variable gain of the saturation element

75

Figure 3.14 Saturation element of example #4

76

Dividing the limits of equation (3.21) by ®. when

©. =10.0,in

£ = 1.0 -^- * 2.0 (3.22)arin

For © greater than 20.0, © = 20.0,a a

©$ = w = i^h

2 % * 20 -°(3 - 23 >

a a

Dividing the numerator and the denominator of the

right-hand side of equation (3.23) by ©. when ©. =10.0.^ ^ in in

20.0"©~~2

©= © /©

n— = ©'/© ^~ ^ 2 *° (3 - 24 )

a / in a / in in

Equations (3.22) and (3.24) define the relationship©

between 3 and q— for a constant ©. equal to 10.0. Thesein

equations are put into tabular form and are shown as table©

(III. 4). The left-hand column is 75:— . Given a certain

e ajs— / P is read from the right-hand column.in ©

With the relationship between -x— and £ established,in

the relationship between j3 and to is now found. Referring©

to figure (3.13), -q-— is written,in

©a _ 14.5s+101.5 /

3 25)in s

4+13s 3 +51s2+63s+14.5+j9(s

3 +10s2+21s)

77

a

in

0.0 1.0

1.0 1.0

2.0 1.0

3.0 0.667

4.0 0.500

5.0 0.400

6.0 0.333

7.0 0.286

8.0 0.250

9.0 0.222

10.0 0.200

Table III. 4 /3 as a Function of -gp— for thein

Saturation Element of Example #4

78

Using the equation (3.23) in the PARAM-5 subprogram, a©

plot of p versus -*;— is made. This plot is figure (3.15)in

and has CO curves of W = 0.1, 0.2, 0.3, 0.4, 0.45, 0.5, 0.55,

0.6, 0.65, 0.7, 0.75, 0.8, 0.9, and 1.0 drawn on it.

With the coordinates of j3 and -g— from table (III. 4),in

the constant ©. curve is drawn on figure (3.15). The

intersection of this curve with the CO curves at the asso-

ciated values of )3, gives the relationship between ft and

CO for a constant ©. = 10.0.mUsing equation (3.19) and the PARAM-7 subprogram, a

®outplot of -s versus CO has been made. This plot is figure

in(3.16) and has curves of £ = 0.0, 0.1, 0.2, 0.3, 0.4,

0.5, 0.6, 0.7, 0.8, 0.9, and 1.0 drawn on it. Using the

coordinates of j3 and co , the constant ©. curve is replot-

ted onto figure (3.16). This constant ©. curve on figure

(3.16) is the closed loop frequency response for example

#4. The range of CO shown in figure (3.16) is only over

the region in which the response deviates from a constant

value of j3. The response at frequencies above CO = 0.7 is

the same as if the saturation element were replaced by a

gain of one. The analog simulation is shown as 'x' on

figure (3 . 16)

.

79

l.J*-

1.2

1.0

H

o.o

Figure 3.15 /3 versus ^r1— magnitude for example #4in

80

Figure 3.16 Closed loop frequency response for example #4

81

F. FIFTH EXAMPLE

The fifth example is a type 2 feedback control system

3for a reproducing contour cutting tool. This system will

be analyzed for the open loop frequency response after

linear compensation has taken place. The nonlinear element

is an ideal relay which has been placed between the compen-

sator and the open loop section of the uncompensated sys-

tem. The block diagram of this system is figure (3.17)

where Ct represents the variable gain of the relay. The

open loop transfer function is,

out 225a(s2+5.6s+7.84) ,_ _ cxS = —

5

2 (3.26)a s (s+5) (s +37.5S+350)

The nonlinear characteristic of the ideal relay in

this example is figure (3.18) where © is the signal going

into the relay and ©, is the signal coming out of the

relay. For every value of © , ®, has only one value,

©, = 40. Referring to figure (3.17), 0i may be written as,

a = -^ © ^ 0.0 (3.27)e

3

Del Toro, V. and Parker, S.R., Principles of ControlSystems Engineering, p. 354, McGraw-Hill, 1960.

82

in in+CO

o «•»*

r-t 4

CO

•°

a

™cE>00 r^

• t

rsj 00+ rH+CO

in «<^

r^ CO

00 (N

Cy

•"

Figure 3.17 Block diagram of example #5 where a representsthe variable gain of the relay

83

O'in

inCM

©°o

i

o

in

i

omi

Figure 3.18 Ideal relay of example #5

84

For ©, = 40. , equation (3.27) becomes

,

« = If^ ©e

* 0.0 (3.28)


hand side of equation (3.28) by ©. when ©. =10.0, eaua-in in M

tion (3.28) becomes,

©<* = ^Q-

^jf-* 0.0 (3.29)

e inw.—in

©Equation (3.29) gives the relationship between 75

—

inand 0L.

From equation (3.29) a table for the relay of a as a©

function of 75— is made up. This is table (III. 5) with© in75— as the value in the left-hand column and the associatedin

value of Ot in the right-hand column. This relationship©

between 0t and 75— defines the constant ©. curve which will©. inm

be used later.©

Because OL is a function of 75— for a constant ©. , a© in

plot of OL versus 75— is needed. Referring to figure (3.17),in ©

the transfer function 75— is written as,in

© 5 4 3 2e _ 4.5s +47.7s +1610s +1768s

in s5+42.5s4+537.5s 3 +1750s

2+a(225s 2+1260s+1765)

(3.30)

85

e

inOL

0.0 M

0.5 8.0

1.0 4.0

1.5 2.67

2.0 2.00

2.5 1.6

3.0 1.33

3.5 1.14

4.0 1.00

4.5 0.89

5.0 0.80

Table III. 5 OL as a Function of -s— for the Relay ofin

Example #5

86

Using equation (3.30) and the PAR/AM-5 subprogram, a

e .

plot of 01 versus -g— is made. This plot is figure (3.19)in

and has drawn on it constant CO curves of to = 0.25, 0.5,

0.75, 1.0, 1.25, 1.75, 2.0, 2.5, 3.0, 4.0, 5.0, 7.5, 20.0

and 40.0.

The constant ©. curve is drawn on figure (3.19) usingin

the coordinates of Qi and -g— from table (III. 5). The in-in

tersection of this constant ©. curve and the CO curves inin

figure (3.19) gives the new relationship between Ot and CO

for a ©. =10.0. With this final relationship, the open

loop frequency response is drawn for the system with the

relay.

From equation (3.26) and the PARAM-7 subprogram, a©

plot of -gj (magnitude - dB and phase) versus CO is made.a

On this plot OL curves are drawn of a = 0.7, 0.85, 1.0, 1.1,

1.25, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 5.0, 6.0, 7.0, 8.0 and©oufc

15.0. The -is versus CO plot is figure (3.20). Note thatin

for different values of OL, the phase response is the same.

This indicates that the nonlinear element does not change

the system from its original linear phase response. The

a = 1.0 curve is the linear open loop magnitude response.

Using the coordinates of 0t and CO, the constant ©. curve3 in

on figure (3.19) is replotted onto figure (3.20). The con-

stant © curves on figure (3.20) are the open loop magni-

tude and phase response of example #5 with an ideal relay.

87

ooNO

I

oOCMH

I

Oo00HI

oo

I

o o oi

Figure 3.20 Open loop frequency response of example #5

89

G. SIXTH EXAMPLE

The system for the sixth example is similar to the

fifth example. The nonlinear element is coulomb friction

plus stiction which converts velocity into torque in a

minor feedback loop. The block diagram of the system is

figure (3.21) where a. represents the coulomb friction plus

stiction element. The open loop transfer function is equa-

tion (3.31) and is taken from figure (3.21).

®out .. 225(s2+5.6s+7.84)

a s5+42.5s

4+537.5s

3+17 50s

2+a(s4+42.5s

3+537.5s

2+1750s)

(3.31)

The coulomb friction with stiction element is shown in

figure (3.22) where © is the signal going into the ele-

ment and T, is the signal coming out of the element. For

less than 2.0, the variable gain can be written as,out

a =4.17 - f © .

3 out

©© < 2.0out (3.32)

out


hand side of equation (3.32) by ©. when ©. = 10.0, equa-


©.417 -$#£

a, = in

©out

W.in

©out n 9

in(3.33)

90

o

to

+CD

LD

•

00

in

•

00rH+00

CO

Figure 3.21 Block diagram of example #6 where a representsthe variable gain of the coulomb friction plus stictionelement

91

.©

ot

m

oo

oin

I

oor-T

i

om

i

Figure 3.22 Coulomb friction plus stiction element ofexample #6

92

For 6>

greater than 2.0, the variable gain can be

written as,

3.3 + .1 ©a =

out©out

© .* 2.0

out (3.34)


hand side of equation (3.34) by ©. when ©. =10.0, equa-


a =

-a -a m 1 _out.33 + .1 -^

in© 77®.out in

©^ * 0.20in

(3.35)

Equations (3 . 33) and (3.34) define the relationship©

between & and outw.in

for a constant ©. = 10.0. . This rela-in

out .tionship is tabulated in table (III. 6) where -g is thein

left-hand column and Ci is the right-hand column.

The transfer function of

at figure (3.21)

.

outw. is written from lookingin

©225s

3+1260s2+1764sout

in s5 +42.5s

4+537.5s

3+1975s

2+1260s+1764 + a(s

4442.5s

3+537.5s

2+1750s)

(3.36)

Using equation (3.36) and the PARAM-5 subprogram, a

^outplot of 0L versus -g is made. This plot is figure (3.23)

inand has constant OJ curves between O) = 0.01 and o> = 10.0

drawn on it. Using the relationship between Ct and -g fromin

93

outw—in

a

.00 «

.05 8.0

.10 3.84

.15 2.45

.20 1.75

.3 1.2

.4 .95

.5 .76

.6 .65

.7 .57

.8 .51

.9 .466

1.0 .43

1.1 .40

1.2 .375

1.3 .354

1.4 .335

1.5 .32

eTable III. 6 a as a Function of q

OU for Coulombin

Friction Plus Stiction in Example #6

94

a

out

inFigure 3.23 Of versus -m magnitude for example #6

in

95

table (III. 6), the constant ©. curve is drawn on figure

(3.23). The intersection of the constant ©. curve withx in

the 0) curves in figure (3.23) gives the relationship between

0L and CO for the coulomb friction plus stiction element.

The final step in the open loop frequency response

outanalysis is making a plot of —g— versus CO . Using equa-

ation (3.31) and the PARAM-7 subprogram, plots of —75— mag-

®out anitude versus CO (figure (3.24)) and —g— - phase versus CO

a(figure (3.25)) are made. Both plots have constant a

curves drawn on them. Using the coordinates of 0L and CO,

the constant ®. curve in figure (3.23) is replotted onto

figures (3.24) and (3.25). The constant ®. curves on

these two plots are the open loop magnitude and phase fre-

quency response of this example. The linear response

(Gt = 0.0) is also shown.

For a better analysis, the closed loop response is

shown in figure (3.26) for the magnitude and figure (3.27)

for the phase. These plots were obtained in the same man-

ner as previous examples. The linear response (Of = 0.0) is

also shown.

96

Figure 3.24 Open loop magnitude frequency response ofexample #6

97

Figure 3.25 Open loop phase frequency response of example#6

98

Figure 3.26 Closed loop magnitude frequency response ofexample #6

99

Figure 3.27 Closed loop phase frequency response ofexample #6

100

IV. THE EFFECT OF CHANGING THE INPUT

A. INTRODUCTION

In the past examples, the frequency response was de-

scribed as a constant ©. curve. The frequency response

was obtained by letting ©. be equal to some specified

value. A change in ©. results in a change in the response.

In this chapter, two examples are analyzed for different

values of input. The first example has a saturation ele-

ment in the forward path, and the second example has a

dead zone element in the major feedback path.

B. THE EFFECT OF CHANGING THE INPUT OF A SYSTEM WITH

A SATURATION ELEMENT

The example of chapter 2 used an input of magnitude

10.0, ©. =10.0. To see what is the effect of various

other amplitudes of signals, ©. of 5.0, 7.5, 10.0, 15.0,

20.0 and 25.0 will be used. The steps in this analysis

are the same as those taken in previous examples. In order

to derive a relationship between gain of the saturation

element and frequency, the transfer function of the signal

going into the saturation element must be obtained. By©

simple manipulation -gp— can be found, as shown in the fol-in

lowing steps:

© = ©. - © (4.1)e in out

101

3k

1—n

—

oiniH+

CNn

o If)

• <r>

o +o CO<£> to

00rH+

-<t

CO

«°

Figure 4.1 Block diagram of servomechanism in chapter 2

102

Dividing by ©. ,^ 2 in

© ee , out-@— - 1 - ^in in

_ -,600a

- 1 T 3 (4.2b)s +18s +95s +150s+600a

s4+18s

3 +95s 2+150s

s +18s +95s +150s+600a

where a is the variable gain of the saturation element.

By using the PARAM program mentioned previously, a

plot of OL, (variable gain of the saturation element) , versus

ethe magnitude of -q-:— is obtained. This is shown in figure

in(4.2). The saturation element may be represented by a vari-

able gain which is a function of the input signal, © .e @

However the plot in figure (4.2) is gain, OL, versus —g— .

inBy the following equations, a relationship between OL and©Q— is developed,in

®-ha = w (4 - 3)

e

where ©, is the signal out of the saturation element as

shown in figure (4.3).

Dividing both the numerator and the denominator of

the right-hand side of equation (4.3) by ©. ,

©V©.

e in

103

ein

Figure 4.2 a versus -m— with constant co curves and constant6. curves inin

104

.0)

Figure 4.3 Saturation element in aervomechanism

105

OL is constant in the region where © ^10.0. In thee

region where © 2: 10.0, ©, , the output of the saturation

element, is 10.0. Thus equation (4.4) becomes,

10.0/©.

° = & /©in

e * 10 -° < 4 - 5 >

e in

For ©. =5.0, equation (4.4) becomesin * *±

©a = 1.0 £- * 2.0 (4.62)

in

©01 = & /& #" * 2 '° ( 4 ' 6b >

e in in

In like manner, for ©. =7.5in

©0L = 1.0 -£— < 1.333 (4.7a)

in

©a = Qy|

3 -£- * 1.333 (4.7b)e in in

For ©. = 10in

©0i = 1.0 -^- * 1.0 (4.8a)

in

©01 = &'/% w~ * 1 -° (4 - 8b)

e/ in in

106

For ©. = 20.0in

a = 1.0 jf- <; 0.50 (4.9a)

©e

©.<;

in

©e

©.2:a = ©°/I ^ * 0.50 (4.9b)

e' in in

For ©. = 15.0in

©0i 1.0 -^- < 0.667 (4.10a)

in

©01 = e /I

7

c" * °* 667 (4.10b)e7 in in

For ©. =25.0in

©0i = 1.0 -^— ^ 0.40 (4.11a)

in

©a = W% W~ * °' 40 (4.11b)

e in in

By using equations (4.6), (4.7), (4.8), (4.9), (4.10)

and (4.11), a table may be made up of |8 as a function of©75— and ©. . This is shown in table (IV. 1).©.in v

'

inUsing the values of OL in table (IV.1) , the constant ©.

in

curves are drawn on figure (4.2), using 0i and 75— asin

coordinates.

107

IX

e©.in

©. =5.0in

©. =7.5in

©. =10.0in

©. =15.0in

©. =20.0in

©. = 25.0in

0.0 1.0 1.0 1.0 1.0 1.0 1.00.1 1.0 1.0 1.0 1.0 1.0 1.00.2 1.0 1.0 1.0 1.0 1.0 1.00.3 1.0 1.0 1.0 1.0 1.0 1.00.4 1.0 1.0 1.0 1.0 1.0 1.00.5 1.0 1.0 1.0 1.0 1.0 0.80.6 1.0 1.0 1.0 1.0 0.835 0.6670.7 1.0 1.0 1.0 0.955 0.715 0.5720.8 1.0 1.0 1.0 0.835 0.625 0.500.9 1.0 1.0 1.0 0.742 0.556 0.4451.0 1.0 1.0 1.0 0.667 0.5 0.41.0 1.0 1.0 0.91 0.606 0.455 0.3641.2 1.0 1.0 0.833 0.556 0.417 0.3341.3 1.0 1.0 0.77 0.514 0.385 0.3081.4 1.0 0.955 0.714 0.477 0.357 0.2861.5 1.0 0.89 0.667 0.445 0.333 0.2671.6 1.0 0.835 0.625 0.417 0.312 0.251.7 1.0 0.785 0.588 0.393 0.294 0.2351.8 1.0 0.741 0.555 0.371 0.278 0.2221.9 1.0 0.702 0.526 0.352 0.263 0.2122.0 1.0 0.669 0.500 0.334 0.250 0.2002.1 0.95 0.633 0.4762.2 0.91 0.606 0.4552.3 0.87 0.55 0.4352.4 0.83 0.536 0.4332.5 0.80 0.534 0.4172.6 0.77 0.514 0.4002.7 0.74 0.4942.8 0.715 0.4762.9 0.69 0.4603.0 0.667 0.445

©Table IV. 1 a as a Function of 75— and ©. for the^©~

in

Saturation Element

108

The intersection of the constant ©. curves and Ui curvesin

gives the relationship between a, the variable gain of the

saturation element, and cc, frequency. It should be noted

that each constant ©. curve intersects a particular con-in ^

stant u.' curve two or three times, giving two or three

values of « for the same frequency. This is the jump re-

sonance effect which will be shown more clearly in the

closed loop frequency response curves.©

Using the PARAMS program again, a plot of -q- versusin

ic is obtained. This plot, figure (4.4), consists of con-

stant # curves of various values between & = 0.25 and

a = 1.0. The coordinates of the constant ©. curves inin

figure (4.2) are a? , frequency, and &, nonlinear gain.

Using these coordinates, the constant ©. curves are re-3 in

plotted in figure (4.4). As was noted previously in figure

(4.2), each constant ©. curve intersects some (£ curves morex ' in

than once. When plotted on figure (4.4), this effect is

clearly seen as being a jump resonance effect caused by the

saturation element.

The effect of changing the input is seen in figure

(4.4). Increasing ©. decreases the magnitude of the re-

sonance peak and decreases the resonance frequency. The

jump resonance remains about the same as ©. changes.

109

ON \0 0"N3 c

c-n

i

Figure 4.4 Closed loop frequency response for various ©.e.

no

C. THE EFFECT OF CHANGING THE INPUT OF A SYSTEM WITH

A DEAD ZONE ELEMENT

The effect of changing the input of the system with a

dead zone element can be seen in the same manner as the pre-

vious example. The block diagram of the system with dead

zone is figure (4.5), where $ is the variable gain of the

dead zone element. The dead zone element is different

from the one used in chapter 2. Its input-output describ-

ing curve is shown in figure (4.6) where © is the signal

out of the element. Note that the non-dead zone portion of

the curve does not have unity gain. The gain is 2.0 when

the signal is not in the dead zone portion.

The frequency of the system will be analyzed for four

different inputs. These inputs are ©. =10.0. ©. = 20.0,^ v in in

©. =40.0 and ©. =80.0. In order to ana-ln in

lyze the problem, the same steps must be taken as in the

preceding examples. The first step is to find a relation-©

ship between qOU and £, the nonlinear variable gain ofin

the saturation element.

Referring to figure (4.5), it is seen that,

©= -&—

out

However j3 = 0.0 when ®out

s 40.0 from figure (4.6). Or,

© = 2(© ,.-40) = 2© .-80. © s 40.0 (4.13a)a v out ' out out

111

JJ

3

jj

3•°

co

oinH+

CNO CO

• in <n.o CTi

o +UD

CO

00r-\

+•<*

•"

rA

Figure 4.5 Block diagram of servomechanism with deadzone element in the feedback path

112

Figure 4.6 Dead zone element of second example

113

2© - 80a outP = ©

out© .

S: 40.out (4.13b)

Dividing both numerator and denominator of the right-

hand sides of equations (4.13a) and (4.13b) by ©in

= 0.0©out < 40

©, ~ ©in in

(4.15a)

2©out

"ST

80©~

t - in in

out in

©out 2 40.0

"ST-

~~©^

in in(4.15b)

Equations (4.15a) and (4.15b) can be solved for each

©. . For ©. =10.0,in in

j3 = 0.0©

in(4.16a)

2©

=

-g^- 8.0in©outWin

©

in(4.16b)

For ©. =20.0,in

= 0.0©out n

in(4.17a)

114

2®out . _—®— " 4 -0 ©

* = W* W& * 2.0 (4.17b)out in^in

For ©. =40.0,in

©= 0.0 ^SS <; 1#0 (4.18a)

in

2® .

out©. "in

2.0 ©

tf* - i.oinout

0.

I = Pp 7T^ * 1.0 (4.18b)

in

For ©. = 80.0,in

pl = 0.0 -^± * 0.5 (4.19a)

in

2®out , n—©— " I- ®

= sP ^ * 0.5 (4.19b)out in

w.—in

Equations (4.16a), (4.16b), (4.17a), (4.17b), (4.18a),©

(4.18b), (4.19a) and (4.19b) give the relation between g,°

inand £ for the various constant ©. . This relation is made

in

into table (IV. 2) where -jg is the enterring value and &in

is read out for a constant ©.in

115

$

eout

in©. =10.in

©. =20.in

©. =40.in

©. = 80in

0.0 0.0 0.0 0.0 0.0

0.5 0.0 0.0 0.0 0.0

1.0 0.0 0.0 0.0 1.0

1.5 0.0 0.0 0.667 1.33

2.0 0.0 0.0 1.00 1.5

2.5 0.0 0.4 1.20 1.6

3.0 0.0 0.667 1.33

3.5 0.0 0.855 1.43

4.0 0.0 1.00 1.5

4.5 0.22 1.12 1.6

5.0 0.4 1.2

5.5 0.546 1.27

6.0 0.667 1.33

6.5 0.77 1.38

7.0 0.856 1.43

7.5 0.93 1.47

8.0 1.00 1.50

8.5 1.06 1.53

9.0 1.11 1.56

9.5 1.16 1.58

10.0 1.20 1.60

10.5 1.24

11.0 1.27

11.5 1.30

12.0 1.33

Table IV. 2 j9 as a Function of®

i.out and ©. forin

the Dead Zone

116

Using the PARAMS -5 subprogram of the PARAMS program,©

a plot of fi versus ^5 with constant O) curves is obtained,in ©

as shown in figure (4.7) . Using the relation between g?u

inand 0, tabulated in table (IV. 2), constant ©. curves are

in

drawn on figure (4.7). The intersection of the constant

©. curves and the w curves in figure (4.7) gives the re-

lation between frequency and £.

The last step in the analysis, is the drawing of the

closed loop frequency response for the various inputs.

Using the PARAMS-7 subprogram of the PARAMS program, a plot©

of qOU

(dB) versus frequency with constant $ curves isin

obtained. This is figure (4.8). Using the relation between

j3 and 03 just found, the constant ©. curves are drawn onJ in

figure (4.8)

.

Figure (4.8) shows the effect of the various inputs.

As ©. is decreased, the magnitude of the flat part of thein r F

response is decreased. At the same time, a jump resonance

effect takes place. This jump resonance is opposite to the

one found for the saturation problem of this chapter.

117

lA

1.2

1.0

$

o.a*• e

f.out

outin

Figure 4.7 j3 versus -s¥— magnitude for second exampleein

118

Figure 4.8 Closed loop frequency response for various Gj.

in a system with dead zonein

119

V. DESIGN TECHNIQUES

A. FIRST DESIGN PROBLEM

The previous chapters explored the techniques of ana-

lysis and showed that they worked for every example at-

tempted. The question is whether or not these techniques

can be reversed. The problem is to start with a given

linear system, and design an element, linear or nonlinear, so

that the system meets a certain specified response. It

must be kept in mind that if a nonlinear element is used,

it must be a signle-valued nonlinearity. To see whether or

not this design can be accomplished, an example will be

worked.

The design problem is a simple third order type 1

feedback system, where the closed loop transfer function is,

© K,Kout 1 2 ,,_ ,v~®—~

s(s+p1)(s+p

2)+K

1K2

^' 1)

as shown in figure (5.1).

Let p, = 1, p 2= 2, and K, = 1.

The first part of the problem is to find a K2

so that

the system is at the stability limit. This is easily done

by conventional methods, and it is found that K2

= 6.

Equation (5.1) now becomes,

out _ 6 (k 2)

in s +3s + 2s + 6

120

,0

Figure 5.1 Type 1 third order feedback system

121

The second part of the design problem is also linear.

The requirement is to provide tachometer feedback in a mi-

nor loop so that M (u>) = 2.0. The block diagram of the

system now is figure (5.2).

As has been previously shown, the PARAMS program can

4solve this aspect of the problem. The transfer function

of the system is now,

©out 6 , .^ - -3 2

(5 * 3)in s + 3s + 2s + 6 + 6K, s

t

where K. is the gain of the tachometer feedback element.

The PARAM-5 subprogram of the PARAMS program provides

an easy method for determining K, when M (to) = 2.0.

Figure (5.3) gives a plot of K. , tachometer feedback gain,© t

versus the magnitude of -zs . Constant it) curves havein

been drawn on the plot for the region of frequency that is

of interest. A vertical line is drawn at 75 = 2.0. Thisin

line represents the desired M (60) . The intersection of any

horizontal line with the constant to curves gives the fre-

quency response of the system. The ordinate value of this

line is K.. Thus the horizontal line at K, = 0.35 givest t

the desired M (a?) =2.0.P

4Glavis, CO., Frequency Response in the ParameterPlane , Master's Thesis, Naval Postgraduate School, 1967.

122

3.0

CO

(No +

• CMV0 CO

ro+

roW

Figure 5.2 Third order feedback system with tackometerfeedback

123

2.1

1.8

1.5

1^

K,

0.0

Figure 5.3 K versus -q

inus J? magnitude for design problem

in

124

The last step in the design is to change M (05) so that

M ( LC ) = 1.2. This is to be done by putting either a non-

linear or linear element in the minor feedback path. This

additional requirement changes the block diagram so that it

now looks like figure (5.4), where & represents the gain

of the nonlinear or linear element. Many techniquess can be

used for designing a compensator to accomplish this new re-

quirement. However in this particular case, a restriction

will be placed on the system. The requirement is that the

compensator be either a constant gain device, i.e., an am-

plifier or attenuator, or a variable gain device, such as a

single-valued nonlinearity.

The next step in this design is similar to the pre-

vious step. A plot of nonlinear or linear gain (or attenu-©

ation) versus -js magnitude (M (co)) is to be made. Thisin p

is done by the PARAM-5 subprogram of the PARAMS program.

The plot is shown as figure (5.5). The ordinate is j3, the

linear or nonlinear gain, which ranges from 0.0 to 2.1.©

The abscissa is -g^— magnitude. A vertical line is drawn© in

at pPu = 1.2. Because the requirements of the problem

instate that M ( to ) = 1.2, no design solution can extend to

P ©the right of the @

out = 1.2 line in figure (5.5). Thisin

region is shaded, and is a 'restricted design zone' as far

as the design is concerned.

The final result in this design is to develop the shape

of an element which, when put in the system for £ in figure

125

JJ

3•°

CO

mw n

O <N •

• +

CO

o

n+ <^.<Dm <£>

w

/^-*v

<j0.

( Vl

®

Figure 5.4 Third order system with linear/nonlinearcompensator in tackometer feedback loop

126

3

o.o

out

in

Figure 5.5 versus q°U magnitude for design problemin

127

(5.4), gives the desired response. Before this can be

obtained, steps similar to those in chapters 2,3 and 4 must

be followed. These steps, however, must be reversed.

Having determined in figure (5.5) the region in which de-

sign cannot take place, a relationship between this region

and the variable j9 is found. The relationship has the co-

ordinates of j3 and frequency. The next step is to find a

relationship between £ and the signal going into the linear

or nonlinear element, using the relationship between /3 and

OJ .

It is difficult to find a relationship between a vari-

able gain, /3, and © in terms of frequency. However, it is

a relatively simple process to find a relationship between©

j3 and -g— in terms of frequency. Looking at figure (5.4) ,

in

©^p— = K

fc

s = 0.35s (5.4)out

Multiplying both sides of equation (5.4) by ©n ,

© = .35s © (5.5)e out

And dividing both sides of equation (5.5) by ©. ,in

© ©e „ _, ,- OUt , r- r \0— = 0.35s -g- (5.6)in in

128

By substituting K = 0.35 into equation (5.3) and

letting the nonlinear or linear element be represented by

ft, equation (5.3) becomes,

eout 6 , ,.@ 3 2 * * '

in s + 3s + 2s + 6 + 2.10s


Gj= ~

2Z (5.8)

in s +3s + 2s +6+ 2.10s

Equation (5.8) is used in the PARAM-5 subprogram to

produce figure (5.6). Figure (5.6) has ft, the variable

gain, as the ordinate and -g—- magnitude as the abscissa.in

The same constant Ui curves that are on figure (5.5) are on

figure (5.6). Using ft and u) as the coordinates for the

restricted design zone in figure (5.5), this zone can be

transferred to figure (5.6). This restricted design zone

represents the area in which design cannot take place.

The line that is the left-hand boundary of the zone is the

critical boundary which will be used to formulate a non-

linear element.

Referring to figure (5.4) again, it is seen that,

(5.9)ft-

a

e

And that,

a= ft®

e(5.10)

129

2.1

1.8

1.5

1.2

3

0.0

Figure 5.6 /3 versus magnitude for the design problem*in

130

Recalling that ©. is assumed a constant, the abscissa

of figure (5.6) is when ©, =1.0. From equation (5.5)t- 1 .'

i

and figure (5.6), the restricted design zone with its left-

hand boundary line can be reconstructed in a plot of

versus © , where © is the signal out of the design ele-e a

ment. This is depicted in figure (5.7) and provides a

ready means of design.

When looking at figure (5.7), a simple solution to the

problem of making M ( U3 ) ^ 1.2 is seen immediately. The so-

lution is to provide a constant gain of 2.0, or j3 = 2.0.

This constant gain is shown as a straight line in figure

(5.7). This solution could have been seen in figure (5.6)

by having a horizontal line at |8 = 2.0. The £ = 2.0 line

in figure (5.6) does not pass through the restricted zone.

This solution is also seen in figure (5.5) as a /3 = 2.0

line also. One could go back in the design process to

where K, was being determined, from figure (5.3), it is

seen that by making K, = .7, a M (03) =1.0 is obtained.

By making j3 = 2.0, one is making a new K, , i.e.,

K.new = $K, for a constant 3.

However, looking at figure (5. 7) again, it is seen

that a nonlinear design is also possible. One such non-

linear design is to have a relay with dead zone. This re-

lay is drawn on figure (5.7). For signals less than 0.25,

the output is 0.0. For signals greater than 0.25, the

131

Relay wit!dead zone

elay withdead zone

Figure 5.7 Plot of © -signal out of the element, versus© -signal into the element, with restricted design zoneand proposed designs

132

output is 1.5. The relay with dead zone describing curve

(constant ©. curve) can be redrawn in the following

manner

:

e= # (5.11)

which comes from figure (5.4) . For this relay with dead

zone,

© = 0.0 © ^ 0.25 (5.12a)a e

© = 1.5 © 2= 0.25 (5.12b)a e v


= -Sf^ © < 0.25 (5.13a)15 e

]." 5 © £W^ e 0.25 (5.13b)

And dividing both the numerator and denominator of the

right-hand side of equation (5.13) by ©. , and letting

©. =1.0 for part of the numerator,in ^

©= 0.0 ^- - 0.25 (5.14a)

in

©* = a/©

5

w~ s °» 25 (5.i4b)e in in

133

From equation (5.14), a table may be constructed simi-

lar to those in the previous chapters. The entering value

is -g— , and the value read off is 3, the nonlinear vari-in ©

able gain. 75— is the abscissa of figure (5.6) and p isin

the ordinate of figure (5.6). The result is table (V.l).

With table (V.l), the constant ®. curve for the relay

can be reconstructed on figure (5.6) . However, since©efigure (5.6) does not cover the range of P and -g— needed,in

a new plot is obtained. This plot is figure (5.8) where 3©

goes from 0.0 to 6.0 and -g— goes from 0.0 to 2.4. It isin

the same plot as figure (5.6), except that the scales have

been expanded. The restricted design zone is reconstructed

on figure (5.8). Now the constant ©. curve for the relay

with dead zone is plotted on figure (5.8). The intersec-

tion of the constant ©. curve and the to curves gives thein ^

relationship between fi and to. Using this relationship in

figure (5.8), the closed loop frequency response can be re-

constructed to show that the relay does meet the specifica-

tions .

The PARAM-7 subprogram of the PARAMS program is used

to reconstruct the closed loop frequency response. This©

plot is figure (5.9) and has -q- magnitude (dB) versusin

03 , with constant 3 curves of 3 = 0.0, 0.5, 0.75, 1.0,

1.25, 1.5, 1.75, 2.0, 2.25, 2.5, 2.75, 3.0, 3.5, 4.0, 5.0

and 6.0 drawn on it. The frequency response for the relay

is drawn on figure (5.9), using the coordinates of 3 and to

134

©e Q

in

0.0 0.0

0.1 0.0

0.2 0.0

0.25 0.0

0.25 6.0

0.3 5.0

0.35 4.28

0.4 3.75

0.45 3.33

0.5 3.00

0.55 2.73

0.6 2.5

0.65 2.31

0.7 2.14

0.75 2.00

0.8 1.87

0.85 1.77

0.9 1.69

0.95 1.58

1.0 1.50

1.1 1.36

1.2 1.25

1.3 1.15

1.4 1.07

1.5 1.00

1.6 0.93

1.7 0.88

1.8 0.83

1.9 0.79

2.0 0.75

©Table V.l j3 as a Function of @— for the Relay with

inDead Zone.

135

Figure 5.9 Closed loop frequency response of designproblem with proposed design solutions

137

from figure (5.8). It is seen in figure (5.9) that the

system with the relay does satisfy the conditions that

M (OJ) < 1.2 (1.58 dB) . Also drawn on figure (5.9) is the

linear design solution, £ = 2.0, which also satisfies the

design conditions. A $= 1 curve is drawn on figure (5.8)

to show the response of the system after determining K.

but before the design of the linear/nonlinear element. The

solution has been checked by analog simulation and the re-

sults are plotted on figure (5.9) by "x's". The analog so-

lution checks closely with the graphical design.

The steps in the design are summarized below. Accord-

ing to the problem, some steps may be eliminated while

other steps may have to be added to solve the particular

problem.

1. After all linear design and compensation have been

completed, with only the nonlinear design left, obtain a

plot of j9, the variable gain of the nonlinear element,@out

versus 75 the magnitude, with constant U) curves drawnin

on the plot. This can be accomplished by using the PARAM-5

subprogram of the PARAMS program or some other similar com-

puter program. The plot corresponding to the example in

this chapter is figure (5.5).

2. Having obtained the plot in step 1, draw on it the

area in which design cannot take place (a restricted design

zone) . This zone is determined by the specifications of

the problem and must be in terms of @ magnitude and fre-in

quency.

138

3. Obtain the transfer function of the signal going

into the nonlinear/linear element with respect to the input

signal into the system. In the problem of this chapter,©

this transfer function is -^— . Using the transfer functionin

ith p* representing the variable gain, obtain a second plotw

of versus this transfer function with the same u) curve as

were drawn in the plot in step 1. This corresponds to

figure (5.6) in this chapter.

4. Using the coordinates of £ and ui from the plot in

step 1, transfer the restricted design zone to the plot ob-

tained in step 3. One now has a relationship for the re-©

stricted design zone in terms of 3 and -^— .

in5. Because ©. must be specified as a constant to ob-

ln ^

tain the response of a system with a nonlinear element, the

plot of step 3 is a relationship between j3 and © for the

restricted design zone. From this relationship, a plot of

© (the signal out of the linear/nonlinear element) versusa

© (the signal into the element) can be drawn. The restric-

ted design zone is then drawn on this plot. This new plot

corresponds to figure (5.7) in this chapter.

6. From this plot in step 5, any linear nonlinear

element can be designed that does not fall into the res-

tricted design zone. The nonlinear element must be a

single-valued nonlinearity, however.©

7. Make a table of P as a function of -g— for thein

element that was designed. This is done by using the plot

139

in step 6, realizing that j3 = -g- and that ®. is ae

constant. This corresponds to table (V.l) in this chap-

ter.

8. Having obtained this table in step 1, plot the

designed element constant ©. curve on the plot of step

5. The coordinates of the constant ©. curve are $ and© m-Q-— and come from the table in step 7. Having plottedin

the constant ©. curve on this plot, one now has the re-in tr i

lationship between |8 and CO for the constant ©, curve.^ in

This relationship comes from the intersection of the de-

signed element's constant ©. curve and the constant COr in

curves

.

9. The final step is the reconstruction of the closed

loop frequency response. The open loop frequency response

can be obtained in the same manner, except by using the

open loop transfer function rather than the closed loop©out

transfer function. Obtain a plot of -@—— versus the fre-in

quency, with constant £ curves. The values of j8 of the

design element should determine what £ curves are needed

in this plot. Replot the constant ©. curve onto this

plot using the coordinates of j9 and CO found in step 8.

The constant ©. curve on this plot should satisfy the de-

sign specifications.

B. SECOND DESIGN PROBLEM

In the previous analysis and design examples, the non-

linearities were functions of the signal into the element.

A second type of nonlinearity is one that is a direct

140

function of frequency. The analysis techniques of this

thesis can be used for this second type of nonlinear ele-

ment. To illustrate how these techniques are used, a de-

sign problem is worked.

The block diagram of the second design problem is

figure (5.10). This system is the same example as was

used : n chapter II. a represents the nonlinear element

which will be used to change the frequency response. From

figure (5.10), the closed loop transfer function is,

out 600a , .

"©"4 3 2 * '

in s + 18s + 95s + 150s + 600a

Using the equation (5.15) and the PARAM-7 subprogram,

a plot of -g (dB) versus 0J is obtained. This plot isin

figure (5.11) and has constant Oi curves plotted on it.

Also on figure (5.11) is the desired frequency response of

the system. The desired frequency response can be describ-

ed in terms of Oi and Ui from figure (5.11).e

From figure (5.10), -§— is written,in

© 4 3 2e s + 1 8s-

3+ 9 5s + 150s .,_ 1CN@— = -7 3 2 (5.16)

in s + 18s J + 95s + 150s + 6000

With equation (5.16) and the PARAM-5 subprogram, a

plot of Qi versus -jap— is made. This plot is figure (5.12)in

and has constant to curves drawn on it. Using the coordi-

nates of Qi and 00 , the desired frequency response can be

141

JJ

3^0©

CO

LOr-H

+CM

CO

ino en

• +o no CO

>£> CO

+

to

®°

a

~o®

\ 1

©

Figure 5.10 Block diagram of second design problem

142

10.0

5.0-

eout

in

0.0

-5.0A .6 .8 1.0 2.

OJ

4.

Figure 5.11 Desired closed loop frequency response ofsecond design problem

143

a

i- e

e inFigure 5.12 a versus -x— for second design problem

in

144

replotted from figure (5.11) onto figure (5.12). As can be

seen in figure (5.11), the desired solution is a functionG

of frequency, not a function of ^— as was the case inin

previous examples. If the solution were a function of&Q

•g£— , then for a given frequency, there would be two valuesin

of gain in figure (5.12). This would not give the proper

frequency response. Figure (5.12) gives the desired solu-

tion in terms of Ot and (0 with jg— as a third parameter.©e ^Ln ©

•jjj-— can only be used as a parameter when -gr— is given within in

an associated value of frequency.

It can be clearly seen from figure (5.12), that for

to £ 1.5, the gain, QL, is 0.4. Between to = 1.5 and 0) = 3.2,

the nonlinear device would have a gain varying bewteen 0.4

and 0.7. For to ^ 3.2, the gain of the element would be 0.7.

This description of the nonlinear device can also be seen

on an input-output plot of the nonlinear element. This dia-

gram is figure (5.13) where a straight line segment repre-

sents the constant gain of the element.

The response of the system is not dependent on ©.

Thus, for this design solution, the frequency response is

the same for all values of ®. . The realization of thein

desired nonlinear element could possibly consist of a vari-

able amplifier with a frequency discriminator to determine

what the gain of the element should be at a certain fre-

quency.

145

2.0"

%in

i.a.

in

Figure 5.13 Characteristics of the nonlinear element of thesecond design problem

146

VI. DISCUSSION, CONCLUSIONS

AND AREAS OF FUTURE STUDY

A. DISCUSSION

The accuracy of the solution of both the analysis and

design depends on graphical accuracy. The plots made by

the digital computer were accurate enough for the study.

Any errors in the solutions came from plotting of the

curves by hand and reading the data from the plots. In

some cases where the scale of the plots was small and ac-

curacy was critical, some small errors appeared in the so-

lution.

All the examples, except for the open loop analysis,

were checked by analog simulation. Appendix A explains the

simulation. The nonlinear elements were simulated by using

electronic switching and digital logic circuits. As a re-

sult, the nonlinear elements are ideal elements. In a real

example, these elements may not be ideal, but instead show

different characteristics. This presents no problem when

using these analysis and design techniques as long as the

nonlinear element remains single-valued. Any single-valued

nonlinearity can be used in the analysis and design.

B. CONCLUSIONS

The main conclusion to be drawn from this study is that

a workable, practical and reliable method has been developed

to obtain the open and closed loop frequency response of

147

systems with a single-valued nonlinearity. This includes

both magnitude and phase response. The method is fast,

accurate, gives a graphical solution and involves no labo-

rious calculations.

The method of analysis can be reversed and be used for

synthesis without complicating the problem. The design

techniques show clearly what are the possible design solu-

tions, linear and nonlinear, including their exact parame-

ters. As in the analysis, the nonlinear elements are re-

stricted to single-valued elements. As shown in the se-

cond design problem, the analysis techniques can also be

reversed to handle design solutions that are a function of

only the frequency of the system. The desired frequency

response remains independent of the magnitude of the signal

going into the system.

The primary problem was finding the relationship be-

tween the nonlinearity and the frequency of the system.

This method provides a very simple technique which permits

the nonlinearity to be expressed as a variable gain and

relates this gain to the frequency of the system simply

and accurately.

The analysis techniques can be used for predicting

jump resonance. The prediction includes not only the fact

that jump resonance will occur but also the frequency of

the jump resonance. This prediction can be made in the

148

first few steps of the analysis. The final frequency re-

sponse shows the jump resonance in both magnitude and

phase.

C. AREAS OF FUTURE STUDY

One of the basic areas that needs investigation is in

the phase domain. With slight modification to the PARAMS

program, phase can be added as one of the variables in

plots other than just the PARAM-7 frequency response plot.

With the addition of phase, an extension of this

method may be possible so that two-valued nonlinearities

could be analyzed. These nonlinearities could include

backlash, negative deficiency, two-position relay, etc.

These types of nonlinearities could also possibly be

studied by breaking them down into real and imaginary

parts and using the two parameter capability of the PARAMS

program to handle these two parts.

Another area left unexplored is analyzing the system

with two single-valued nonlinearities. This could be de-

veloped as follows. The last step in the analysis tech-

nique was replotting the constant ©. curve to a plot of© in

q° versus CO to obtain the frequency response. Insteadin

of doing this replotting, transfer the constant ©. curve© in

to a plot of the nonlinear gain versus -g . In thein

PARAMS program, some element had to be designated as an-

other parameter value and held constant for the analysis.

®outBy making a series of plots of nonlinear gain versus -^

in

149

with the constant ®. curve, each plot having the second

parameter set to a different value, a three dimensional

surface of the constant ©. curve is developed. The coor-ln r

dinates would be Ot, the nonlinear gain, 0, the second pa-

®outrameter, -q- and to . By repeating this with a second non-

inlinear element, a second constant ©. surface is obtained.

in

The intersection of these two surfaces at a certain to and

-Q- gives the closed loop frequency response of the sys-in

tern with two nonlinear elements.

This technique just described would be of course very

laborious. It could be greatly simplified by developing

a digital computer program which would find the surfaces

and their intersections. This could then be extended into

a four dimensional space program to handle three nonlinear-

ities, five dimensional space to handle four nonlinearities,

etc.

150

APPENDIX A. ANALOG SIMULATION

The analog simulation was performed on the

Donner 10/20 analog computer and two EAI TR-20 analog com-

puters. The Donner 10/20 was used to simulate the linear

portion of the system and the two EAI TR-20s were used to

simulate the nonlinear element.

The simulation of the linear portion of a system was

done by conventional analog simulation techniques. The

main consideration was to preserve the position of the non-

linear element in the circuit so that no accidental mani-

pulation occurred. This manipulation would correspond to

incorrectly manipulating a block diagram with a nonlinear

element. The reason for this care is the fact that the

response of a system is a function of where the nonlinear-

ity is placed in the system.

Scaling had to be used throughout the simulation and

particularly in using the Donner with the two EAI computers.

The Donner is a 100 v. machine and the EAIs are a 10 v.

machines

.

The success of the total simulation rested with the

nonlinear element simulation. For this two EAI TR-20

electronic comparators were used, one electronic compara-

tor in each machine. Each electronic comparator consists

of a comparator unit which produces binary outputs and

two electronic switching units. The comparator and switches

151

have a switching time of 1.0 fl sec. and the comparator has

a switching sensitivity of ± 1.0 jlv. The block diagram of

5one electronic comparator is shown in figure (A.l).

The electronic comparator can be patched so that it

compares two inputs, X and Y, so that,

X + Y * 0.0 E = E, (A. la)o 1

X + Y < 0.0 E = E2

(A. lb)

Where E, and E~ are two arbitrary inputs and E is the

output of the electronic comparator.

One electronic comparator was used for positive sig-

nals and the other electronic comparator was used for ne-

gative signals. Two examples are worked to show how the

units were used.

The first example is an ideal relay (such a figure

(3.18)) where it is desired that for all positive signals

into the relay, the output is 4.0 volts. For all negative

signals into the relay, the output is -4.0 volts. For the

first electronic comparator, X is the input signal, Y is

zero, E, is 4.0 volts and E?

is the input signal. Thus,

when X + Y ^ 0.0, which is when the input signal is great-

er than zero, the output from the first electronic com-

parator is E, or 4.0 volts. When X + Y < 0.0, which is

5

Electronic Associates, Inc. TR-20 Computer Opera-tor's Reference Handbook, 1964.

152

Figure A.l EAI TR-20 electronic comparator

153

when the input signal is negative, the output of the com-

parator is Ep or the negative portion of the input signal.

The output, E , of the first electronic comparator is made

the X and the E-, of the second electronic comparator. The

Y is zero and E~ is -4.0 volts. Thus when the output of

the first electronic comparator is negative, the output of

the second electronic comparator is -4.0 volts. When the

output of the first electronic comparator is positive, the

output of the second electronic comparator is +4.0 volts

which is the only positive output of the first electronic

comparator.

The second example is a dead zone element whose dead

space extends ±4.0 v. Let the input into the dead zone

element be represented by the symbol IN. For the first

electronic comparator X = IN, Y = 4.0 v., E, = IN - 4.

and E9

= 0.0. Thus for positive signals greater than +4.0

v., the output, Eni , of the first electronic comparator is

IN-4. For signals less than +4., EQ

, = 0.0. For the

second electronic comparator let X = IN, Y = +4., E, = E„,

and E~ = IN + 4. Thus for signals less than -4., the

output, En? , is IN + 4 . For signals greater than -4.,

E02= E01 which is zer° for "4 - * IN * 4 -° and IN-4.

for IN > 4.

With the switching time of l.Ojisec. and sensitivity of

lfXv. , the response is very good. The output of the second

electronic comparator is fed back into the linear system

with appropriate scaling.

154

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174

LIST OF REFERENCES

1. Del Toro, V. and Parker, S.R., Principles of ControlEngineering , McGraw-Hill Book Company, 1960.

2. Electronic Associates, Inc., TR-20 Computer Operator'sReference Handbook , 1964.

3. Glavis, CO., Frequency Response in the ParameterPlane , M.S. Thesis, Naval Postgraduate School, Monterey,California, 1968.

4. Ruivo, F.M.P., Frequency Response Study of NonlinearNetworks by Parameter Plane Techniques , M.S. Thesis,Naval Postgraduate School, Monterey, California, 1968.

5. Staples, D.S., Frequency Response Analysis of NonlinearDynamic Sys terns , M.S. Thesis, Naval Postgraduate School,Monterey, California, 1968.

6. Thaler, G.J. and Pastel, M.P., Analysis and Design ofNonlinear Feedback Control Systems , McGraw-Hill BookCompany, 1962.

175

INITIAL DISTRIBUTION LIST

No. Copies

1. Defense Documentation Center 20Cameron StationAlexandria, Virginia 22314

2. Library, Code 0212 2

Naval Postgraduate SchoolMonterey, California 93940

3

.

Commander 1Naval Ordnance Systems CommandHqs . Department of the NavyWashington, D.C. 20360

4. Professor G. J. Thaler 10Department of Electrical EngineeringNaval Postgraduate SchoolMonterey, California 93940

5. Dr. D. D. Siljak 1Department of Electrical EngineeringUniversity of Santa ClaraSanta Clara, California 93153

6. Mr. J. Karmarkar 1

Department of Electrical EngineeringUniversity of Santa ClaraSanta Clara, California 93153

7. Dr. A. G. Thompson 1

Department of Mechanical EngineeringUniversity of AdelaideAdelaide, SOUTH AUSTRALIA

8. Dr. J. Peracaula 1

Division of Automatic ControlTechnical UniversityJ. Gutierrez Abascal 2

Madrid, 6, SPAIN

9. Dr. M. Rakic 1

Marsala Tita 6

Belgrade, YUGOSLAVIA

10. Cap t. R. D. Staples, USMC 1

1816 Ridge RoadEl Dorado, Kansas 67042

176

No. Copies

11. Lt. F. M. P. Ruivo 1

Minis terio da MarinaPortuguese NavyLisbon, PORTUGAL

12. Dr. Lawrence Eisenberg 1

Department of Electrical EngineeringMoore School of EngineeringUniversity of PennsylvaniaPhiladelphia, Pennsylvania 19104

13. Lt.JG John P. Davis, USN 1

173 Little Neck RoadCenterport, New York 11721

177

UnclassifiedSecurity Classification

DOCUMENT CONTROL DATA R&D(Security classification of till*, body of abstract and indexing annotation must be entered when the overall report la classified

I originating AC Tl vt T Y (Corporate author)


2a. REPORT SECURITY C L A SSI F I C A T I Or

Unclassified2b. GROUP

3 REPORT TITLE

Frequency Response Analysis and Design of Single-Valued NonlinearSystems Using the Parameter Plane

4. DESCRIPTIVE NO TES (Type of report and, inclusive dates)

Master's Thesis; June 19695 authoR(S) (First name, middle Initial, last name)

John P. DAVIS

6- REPORT DATE

June 1969

7«. TOTAL NO. OF PAGES

17976. NO. OF REF3

»a. CONTRACT OR GRANT NO.

b. PROJEC T NO.

9«. ORIGINATOR'S REPORT NUMBER(S)

9b. OTHER REPORT noisi (Any other numbers that may be assignedthis report)

10. DISTRIBUTION STATEMENT

Distribution of this document is unlimited.

II. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY


13. ABSTRACT

Frequency response techniques are a valuable tool in the analysisand synthesis of linear systems. Extension of these techniques is madeto analyze and design systems with a single-valued nonlinear element.The relationship between the characteristics of a nonlinear element andthe frequency of the system is developed by simple calculations and a

digital computer program.

DD F0«" 1473I NOV »o I *T # w

S/N 0101 -807-681 1

(PAGE 1)UnclassifiedSecurity Classification

1-31408

179

UnclassifiedSecurity Classification

key wo R DS

Single-Valued Nonlinear Systems

Parameter Plane

Frequency Response Analysis

DD FORM.1473 < BACK >

S/N 0101 -807-6821Unclassified

180Security Classification A- 3 1 409

floraSHELF BINDER

i Syracuse, N. Y.

; Stockton, Calif.

thesD1715

Frequency response analysis and design o

3 2768 001 02334 4

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