the equalization and optimization of a third-order type-1
TRANSCRIPT
Scholars' Mine Scholars' Mine
Masters Theses Student Theses and Dissertations
1960
The equalization and optimization of a third-order type-1 position The equalization and optimization of a third-order type-1 position
control system control system
Viswanatha Seshadri
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Recommended Citation Recommended Citation Seshadri, Viswanatha, "The equalization and optimization of a third-order type-1 position control system" (1960). Masters Theses. 2800. https://scholarsmine.mst.edu/masters_theses/2800
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THE EQUALIZATION AND OPTIMIZATION
OF A
THIRD-ORDER TYPE-l·POSITION CONTROL SYSTEM
BY
VISWANATHiL:, SESHADRI
A
THESIS
subm1tted to the faculty of the
SCHOOL OF MINES AND METALLURGY OF THE UNIVERSITY OF MISSOURI . . '
in partial fulfillment of the work required for the
Degree of . ~t MINES
MASTER OF SCIENCE IN ELECTRICAL ENGINEERING ~~y:'\i. ·-,~, ·t· . . - . ' 2;'(/ •'' \:,-·:<\ '. (Q .... \ ~
· Rolla, Missouri .· ,.,J/ \..;·:>;; 11 I q._,0'\J ~
. ~ I · "b '1. "\S . ~ I x /\ '<)
1960 ~;''.?, \ t, r-~',\ )., .. ·, "I • ~_;: \ "" v \ \ \ /
... .::::. . :'. :· ·. t :-~/ ~?·
/;) . /L / 1 Approved by )~ /1 ..._ L) ~Li,~· ~U)<r014£2 { 'j/u1fi:cadvisor)~/Jn~ -
U ~A/~~--- . · ... .. . .... .... . . .... .
lidb7k>0~ . · ··.e. c /([). !fL2s~Lc. . , , . ~ ... .. . _._ ' (' ·.··· .. ~ ... , . ', .· · . . · .. :
ACKNOWLEDGEMENTS
The author wishes to thank the Chairman and
members of the faculty of the Department of Electrical
Engineering for the laboratory facilities fhat made
this work possible and for the friendly encouragement
that rendered the work pleasant.
The author is particularly indebted to Dr. R.
Nolte and Professor R. T. Dewoody for their valuable
guidance and advice both in carrying out the investi
gation and in preparing this report.
The author is also grateful to the Director and
staff of the campus Computer Center whose help and co
operation made possible the large volume of calculations
on which much of the analytical and graphical parts of
this work are based.
Thanks are also due to Mr. Jack Boyd of the
English Department who was of considerable help in
organizing the .preparation of this report and to
Mrs. Anna Lee Vannoy who did the typing.
TABLE OF CONTENTS
Page
Acknowledgements DD ODO DO O ODO O O .o O •.• 0 0 DD DO O O O O O O a DODD DO O ODO O ~
Table of Contents O. 0. DOD. D •• a ••• a •• a .D a a.· 0 ••••••• 0 0 0 •• 0 0 0 0 3
List Of Tables ~ 0 • 0 0 0 0 a • 0 D •.• a O D D O D • 0 • 0 ••• 0 D O •• 0 ia • 0 D O • ~ •••• 4
List of Figures a ••••• · D .. 0 0 • ~ 0 0 a O • D • 0. 0 •• 0 D D D a ••••• D •••• 0 0 • 5
List of Symbols ••••••••••••••••••••••••••• · ••••••••••••••• 7
r· 0 Introduction. D O •••••••••••• D O O •• ~ ••• D •• 0 D ••• ~ ••••••• 11
II. Review of Literature •••••••••••••••••••••••••• ~ •••• ~u
III. Choice of Basic System ........................... · •• 34
IV. Design of Phase-advancement ................ ·• •••••••• 46
V. Design of Gain~advancement •••• ~ •••••••• ~ •••••••••••• 65
VI. Non-linear Compensation •••••••••••••••••••••••••••• 94
VII. Conclusions •••••••••••••••••••••••••••••••••••••• 111
Appendix I •••••••••••••••••••• ~ •••••••••••••••••••••••• 116
.Appendix II. 0 DO. 0 0 ••• 0. 0 0 0. a •• 0 a • 0 0 0 0. 0 0 0 D ,•DD O O O a ODDO OD 163
Appendix III •••••••••••••••••••••••••••••••••••••••••• ~1~5
Appendix IV ••••••••••••••••••••• ~ •••••• ~ ••••••••••••••• 130
Bibliography a a • D O O • 0 0 D •• 0 0 D • D a a D O O • a O D D a a a a .. 0 D O • D O •• . D •• 13 5
Vi ta D D a a DD D • D • D •• D O ~ D DD O a 1t O O O D D ·• DD O •••• 0 D • 0 0 • D ••• ~ •• a • D 14i
4
LIST OF TABLES Page
I. Lag Network and Transient Performance ••••••••••••••• 73
II. _Lag Network and System Characteristics •••• .•••••••••• 84
III. Transient Performances of the Low-, High- and the Variable-Gain Systems •••••••••••••••••••••••••• 100
IV. Transient Performance of the Variable-Lag System ••• 106
v. Transient Performance of the System Under Changes in Lead Network••••••••••••••••••••••••••••••••••••l08
2o
5
LIST OF FIGURES
Page Common Form of a Simple Position Control Systemoooo35
Block Diagram of Basic Position Control Systemooooo38
3o(a) Block-Diagram of Basic System Showing the Location of Load Disturbanceso (in terms of poles)ooooooooo39
Jo(b) Block Diagram of Basic System Showing the Location 9f Load Disturbanceso (in terms of time-constants)40
9o
lOo
llo
120
130
140
16.
Frequency-Respon~e and rhase-Angle of Basic Systemo42
Root Locus of Basic Systemooooo~ooooooooo~ooooooooo43
A Simple_Form of Lead Networkoooooooooooooooooooooo47
Analogue Simulation of a Simple Lead Networkooooooo49
Phase-Advancement with a Lead Networkoooooooooooooo50 -
Maximum Phase-Advancement from a Lead Networkoooooo52
System Gaip. of ·uncompensated System for Different · Values of aain-Cross-Over Frequencyoooooooooooooooo55 ,
Phase-Margin of Uncompensated Third Order Type-1 System for Different Values of Gain-Cross-Over Frequency o o o o o o o o o o o o o o o o o o o o o·o o o o o o o o o o o o o o o o o o o o o 58
Phase-Margin of Phase-Compensated Third-Order Type-1 System for Different Values of Lead Network Ratio ·and Gain-Cross-Over Frequencyoooooooooooooooo61
A .Simple .Form of Lag Networkooooooooooooo ·ooooo_ooooo66
Analogue· Simulation of .a Simple Gain-Advancing. Systemo~ooooooooooooooooooooooooooooooooooooooooooo67
Analogue Setup 0£ Compensated System with Variable Design of .. Lag Network o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 70
(.$).tof)Input-. and Load-Disturbed Transients for Different Positions of Lag .Network on the Frequency Scaleoo74-5
6
17. Page (atof) Frequency-Response and Phase-Angle of the
Compensated System for Different Positions of Lag Network on the Frequency-Scale ••••••••••••••• 76-81
180. . Polar Plot of Frequency Response and Phase.-Angle of the Compensated System for Different Positions of Lag Network on the Frequency Scale ••••••••••••••• 82
190 Root Locus of the Compensated System for Different Positions of Lag Network on the Frequency Scale ••••• 83
20. Typical Placement of Poles and Zeros in a Compensated Third-Order Type-1 System ••••••••••••••• 91
210 Input-Step-~ransients of Low and High-Gain Systems •• 96
22. Analogue _Setup of Variable-Gain System ••••••• ~ •••••• 98
23. Input-Step-Transients of Low-, High- and Variable-Gain Systems.•••••••••••••••••••••••••••••••••••••••99
24. Analogue Setup of System with Variable Lag Networi.104
25. Input-Step-Transients of Highly-Damped, Lowly-Damped and Variably-Damped Systems, Obtained by Changing Lag Network•••••••••••••••••••••••••••••••l05
A-1. Schematic Diagram of Servo-Motor and Load •••••••••• 117
A-2. Flow chart of Computations in Appendix III ••••••••• 129
A-3. Flow chart of Computations in Appendix rv •••••••••• 134
Symbol
a
c
e
E
f . m ¢
¢(max)
7
LIST OF SYMBOLS
What it Represents
Characteristic ratio of an equalizing network · { a<l in a lead network; a;>l in a lag network; a= z/p) ·
Capacitor in an equalizing network.
Damping Ratio of the least-damped roots of a system.
Error signal in a servo-system.
Steady-state position error.
The error function that is minimized for system optimization.
Voltage applied across the armature of a servo-motor.
Controlled signal from a control system.
Input signal to a transfer function or element.
Output signal from a transfer function or element.
Reference signal to a control system.
Viscous friction of the load on the servomotor.
Effective viscous friction of motor and load with reference to load.
Viscous friction of servo~motor.
Phase advancement of a lead network.
Maximum phase-advancement by a lead network of given ratio •
. Phase margin.
¢m(max.)
G
Qt
gm
H
r\ . ,,t ~c· · a
K
Maximum phase margin for given lead network ratio and given cross-over frequency.
General form of the operational part of a transfer function, in terms of system time-constants.
· General form of the operational part of a transfer function, in terms of the poles and zeros of the function.
Gain margin
Feedback transfer function.
Controlled value of angular position or, the output.
Reference value of angular position or, the input.
Armature current in servo-motoro
Field current in servo-motor.
Polar moment of inertia of servo-motoro
Polar moment of inertia of load on servomotor.
Effective polar moment of inertia or servomotor and load with reference to load.
The constant part of the general form of transfer function in terms of time-constants.
The constant part of the general form of transfer function in terms of poles and zeros.
Gain constant of amplidyne.
Amplification of difference amplifier.
Back-emf constant of servo-motor.
Gain-constant of servo-motor.
Gain-constant of the uncompensated system, at wcg = wcp•
L
L n
m
N
T
System gain of uncompensated system, at wcg = wcp•
Potentiometer transfer constant.
Torque constant of servo-motor.
Velocity error constant. (Kv: Kin a type-1 system). . .
External load torque on position contr.ol system.
Normalized external load torque on position control system.
Slope of a straight line.
Ratio of reduction gearing from servomotor to output abaft~
Differential operator.
Smaller non-zero pole of basic third order system of type 1.
Larger non-zero pole of basic third order system of type 1.
Pole introduced by gain-advancing network.
Pole introduced by phase-advancing network.
Resistance of armature of servo-motor.
Resistances in equalizing network.
Ratio of gain-cross-over fr~quency to phase-cross-over frequency.
Time constant corresponding to the zero of an equalizing network.
9
v w
Time constant of amplidyne.
Time constant of servo-motor.
Rise time, upto final value, of inputstep-transient of a system.
Rise time, upto 63% of final value, of input step-transient.
Time-constant of decay of load-disturbed transients
S!ttling time, within 5% of final value, of input-step-transient.
Constant velocity of ramp input.
Frequency in radians per sec.
Gain-cross-over frequency.
Asymptotic gain-cross-over frequency
Phase-cross-over frequency.
Undamped natural frequency of a system.
Frequency of damped oscillations in a system.
Zero introduced by gain-advancing system.
Zero introduced by phase-advancing system.
10
11
I. INTRODUCTION.
While the practice of feedback control dates back to
the day when the earliest living organisms began .to react to
their environmental conditions and the changes therein,
it has been only during the past two decades that this
technique came to be subjected to a scientific examination.
Yet, an astoundingly rapid advancement has resulted in the
field of servo-mechanisms, contributing substantially to
the development . of modern civilized existence and activity
to its present standards, and promising to contribute, to
an ever-increasing extent, to the accelerated progress of
scientific and industrial enterprise in this space age.
The principal concerns of the theory of servo
mechanisms are two-fold:
(i). The analysis and evaluation of the
performance of a given feedback control system, on the
basis of a knowledge of the characteristics of the com
ponents comprising the forward and the feedback loops;
{ii). The synthesis and design of the necessary
components of the forward and the feedback loops, in order
that the over-all closed loop control system conforms, in
its performance, to certain requirements dictated by the
function and purpose of the particular control system.
A variety of inter-related methods have been developed to
serve the above ends.
12
A ve-ry common and relatively simple .form of servo
mechanism is the position control system, where the output
is determined by ~ ref erenc·e· input o Whenever the reference
value ~changes, the output has to follow the same changeso
Also_, if ~he output value is disturbed, for some reason or
other, . it has to be regulated back to .be equal to the
reference valueo The design of such a follower-cum
regulator system normally requires a type-1 system~ so
that zero steady-state position error may be obtained
under a unit-st.ep position input o Such a system consists,
in its common form, of a variable-voltage generator supplying
a variable-torque motor,· which, together, constitute a
basic third order systemo
Such a third order type-1 system normally requires
compensation for stability by phase advancement at the
~perating frequencies of the system, either by means of a
lead network, or with unity-cum-rate feedbacko It is also,
in general, necessary to increase the low frequency gain
appr~ciably in order to conform to the bounds. on steady
state error under unit-rate inputo A lag network, with
relatively low corner frequencies, provides substantial
attenuation at higher frequencies and thus permits an
increase in overall system gain without appreciably changing
the operating conditions and stability characteristics
of the systemo
13
The unit-step function incorporates a wide range
of frequencieso The highest frequencies are represented.by
the steep initial rise, while the lowest frequencies account
for the subs~quent flat and constant-value regiono Hence,
a study of the response of a system to a unit-step dis
turbance . yields information concerning its response for_ a
wide range of frequencies of sinusoidal disturbances, in
addition to giving the practical information regarding
system behaviour under severe disturbanceso It is thus
possible to interpret the nature of the transient response
of the closed loop system to a unit-step input ·disturbance
in terms of its frequency response and its stability and
performance characteristicso Procedures have also been
developed to obtain the nature of closed loop transient
response directly from the transfer functions of the
forward and the feedback loopso
In designing the networks for phase advancement
at the operating frequencies and for increase in gain at
low frequencies ., the objective is to obtain some degree of
equalization between the conflicting requirements dictated
-by the.need for good stability.on the orie hand and the
need for low steady-state error under unit rate input on
the othero But~ it is often possible and necessary to
design the equalizing net't"rorks in such a way as·:: to get the
14
. optiraum ·system giving the best forra of transient respons·e
to unit-step disturbanceo Several criteria have been
suggested for the determination of the optimum transient . .
response, 'tlhich include mathematical and experimental
methodso
In the _design of equalizing networks for a follower
cum-regulator type system~ the. process of optimization
·has to take into consideration the transient responses of
the output func~ion for t~o kinds of _possible disturbances:
(i)o a unit-step disturbance applied at
the input;
(ii)-.o an equivalent uriit-step disturbance . .~
applied at the point of ~ontrolo The designer could then
be assured of equally good performance ·of -the system
both as .. a follot1er and as a regulatoro
It is a purpos~ or this project to ca~efully examine,
by analogue and digital computer techniques, the effects
of cha~g~s in the design of the phase-advancing. and ~he
gain-advancing ·sections or the equalizing system on the .. .
transient responses of the over-all ·system ~o input and
load disturbanceso
?lhile the criteria of design ,trl.th reference to
optimiz~tion of transient response to ~nput disturbance
appear to have ·received considerable attention and
discussion, an equivalent attention has not been given
to the need for ensuring a good transient response to
load disturbanceso But, the importance of the latter
requirement, particularly in regulating systems, is
easily app~eciatedo
15
·While conventional .design procedure offers many
criteria for the selection of -the lead network functions,
it appears to leave the choice of the lag network rather
arbitraryo But., it is also generally knotm that the .
choice of the lag network is restricted to a relatively
small region for its upper corne~ frequencyo The region
itself is determined by the other functions of the systemo
A substantial departure from the region, either way,
results in a considerable deterioration in the transient
responses of the systemo It is proposed to study these
variations and arrive at an empirical relationship con-
_necting the upper corner frequency of the optimum lag
network to the other corner frequencies and the· gain
cross-over frequency of the systemo
It has .also been recognized that the . zeros of
the open loop function, smaller in value than the gain
cross-over f~equency of the system, emerge as those poles
of the closed loop system w~ich determine the nature of
·settlement of the closed loop transientso The zero of
16
the lag network thus has a significant effect on tran
sient settlemento A study is hence made of the effects ·
of changing the zero of the lag network on the nature
of settlement of the transient responses of the system
to input and load disturbanceso Based on the results,
certain possibilities are brought out for predicting
restrictions on the position of the lag network zero on
the frequency scale on the basis of the settlement require
ments on input and load disturbed. transients.
It is a common feature of servo-mechanfcal optimi
zation that a compromise has to be made between the high
gain required for low error in the steady state and the
relatively low gain dictated by the good stability
requiremento The lag network represents this compromise,
since it provides high gain at the low frequencies obtaining
at the steady state and a lower gain at the comparatively
higher frequencies of system resonation under transient
conditionso
·The designer also makes another. significant com
promise with regard to optimization. It is generally
recognized that the rise time of transient respons·e to a
step input is nearly inversely proportional to the system
band width, represented by the gain-cross-over frequency
of its open loop function. Though a high cross-o.ver
17
frequency is thus desirable, the design of phase advance
ment for reasonable values of phase margin and damping
ratio impose a limitation on the maximum possible value
for cross-over frequency. A compromise is thus made,
in the choice of the cross-over frequency of the open
loop function, between the con.flicting requirements of
low rise time and good damping.
Several alternatives have been attempted for over
coming the above conflicts of design criteria and hence to
avoid compromises in optimization. Mathematical approaches
have been suggested for the determination of the open
loop function, which includes the forward and feedback
paths, for any arbitrary closed loop performance and tran
sient required. Such functions may often have to be non
linear in their characteristics of frequency-response as
well as amplitude response. While any arbitrary frequency
response may be obtained with the proper choice of linear
circuitry, the methods of non-linear function generation
must be employed in order to obtain the necessary non
linear amplitude response pattern.
The techniques of non-linear circuitry appear to
have developed mainly as a result of attempts to obtain
electrical analogues of the common non-linearities such
as dead space, backlash and saturation encountered in the
electro-mechanical components generally employed in servo
mechanical systems. These have been or substantial assist
ance in the analogue computation of the performance of such
systems. Since the supercession of the relay circuits by
the biased-diode type of non-linear elements, the generation
of complex non-linear functions has become easier and
possible to a high degree of fidelity.
Various methods have been suggested for the design
of open-loop functions having non-linear amplitude response.
But, most of such methods appear to involve a considerable
degree or complexity, and consequently a large cost of
design and assembly, which may not often be warranted
for the optimization of relatively simple systems. In
such systems, an extremely high degree of optimization
would be unnecessary and uneconomical. However, any
simple method of providing a substantial improvem:ent in
optimization at small additional cost would be ideally
suited for such systems.
A study. is, therefore, made of the possibilities
of substantially improving, with a few simple nonlinearities,
the optimization of transi~nt response of the simple
type-1 third order system equalized with linear circuitry.
The inve·stigation is done by analogue computation techniques.
The extent of non-linear circuitry is limited to a small
number or biased-diode loops. The purpose of the study
is to find the best among the simplest of non-linear
arrangements that would give the system a low time of
recovery from large errors along with good damping and
low steady state error after the error is reduced to
a relatively small value. Three possibilities are
studied where the non-linearity introduces a change
in: (i) gain, (ii) the lag network, and (iii) the
lead network. Conclusions are drawn concerning the
most desirable design for the purpose, considering the
simplicity of circuitry and the degree of improvement
in optimization possibleo
19
20
rr. REVIEW ·oF LITERATURE°.
The theory and practi~e of the analysis and ·synthesis
of servo-mechanisms.have been developed largely over the
last two decades. The earlier developments in the fields
. of electrical machines, electronics, communications and
instrumentation made it possible to draw out schemes for
dependable automatic feedback controls for the -many indus
trial, military or other applications of the day. The
theory of servo-mechanisms concerns itself with the deter
mination of the performance of s~ch a·control system and
explores the possibilities of improving it with regard to
error and stability.
A. Analysis and Synthesis of Linear Systems:
A linear ~lectro-mechanical srstem is one in which
the dynamic variable undergoes only pure integration,
differentiation or multiplication by a constanto The
dynamics of s~c.h a system forms the solution of a linear
differential equationo
A knowledge of the system differential equation
permits the evaluation of system performance under any kind
of input at any pointo S~veral mathematical methods,
including the.Laplace transformation, are available for
this purposeo
21
·It is convenient to obtain the dynamic characteristic
of each component of the system in the form of a transfer . .
function in terms of the differential operatoro These
could then ?e manipulated algebraically, and th~ over
all . dynamic behaviour of the system may be evaluated by
inverse Laplace transformationo
The determination of the operational transfer
function of an electro-mechanical element can be accomplished
in several wayso The basic dynamics could be expressed
mathematically and the operational function derived
therefromo This method . is useful for simple systems
whose eledtro-mechanical parameters are accurately knowno
The 'frequency-response' method takes advantage of
the special ma~~ematical properties of the sinusoid,
whereby a differentiation amounts to amplification by . .
a factor equal to the fre~uency and an in~egration amounts
to attenuation by a factor equal to the frequencyo Hence,
a semilogarithmic plot of the frequency-response of.the . .
sy~tem for siriuso~~a~ inputs provides information con
cerning the corner frequencies of the factored transfer
function a~on~ with the multi~licity or ~he complexity
at each cornero These same principles are employed in
making the 'Bode plot', or an asymptotic approx~mation
to frequenc; response··'rrom the factored transfer functiono
22
Other methods have been suggested for the determination
of transfer functions. The 'Self-oscillation method',
described by Clegg and Harris (1), where' the system is
allowed to oscillate and supply its own driving force'
is .worth mentioning. The 'High-power Servo-analyser' (2),
described by Gorril and Walley determines 'the frequency
response curves, both magnitudes and phase-shifts, of
actual apparatus operating at normal power levels'. Lendaris
and Smith (3) have developed a 'complex-zero signal generator'
useful for determining the characteristic poles of a system.
The signal is fed into a system to cancel the existing poles.
The cancellation is judged by a null method and the complex
pole cancelled is read on the calibrated zero-generator.
The open loop transfer function, consisting of the
forward and the feedback loops, is thus obtained. But-,
obtaining the inverse Laplace transformation of the closed
loop transfer function is often dj_fficult. It involves
the laborious process of reducing it to such simple partial
fractions for which the inverse Laplace transformation is
either available or easily determined. This happens to be
the major hurdle in the process of servo-mechanical analysis
and synthesis. Many methods have been developed for over
coming this difficulty and to arrive at the performance of
the closed loop system directly from the open loop transfer
function.
(1) All references are given in the bibliography.
23
The frequency-response of the open loop function,
a semi-logarithmic plot of which is easily made by Bode's
method of asymptotic approximations, has been shown to
offer a variety of clues concerning the nature of the
closed loop performance of the system. The gain-cross
over-frequency, wcg, at which the frequency-response curve
crosses the zero db axis, and the magnitude of the open
loop function becomes unity, together with the phase-cross
over-frequency, Wcp, at which the phase angle of the open
loop function becomes 180 degrees, yield the order of
magnitude of the frequency of damped oscillations, wr,
of the system.
At this frequency, the denominator of the closed
loop function attains minimum magnitude and the magnitude
of the closed loop function itself becomes a maximum. This
maximum value of the closed loop function,~, has been
recognized as an indication of the degree of damping of the
system, a higher value implying a low degree of damping.
A simple relati~nship exists between~ and the damping
ratio in a second order system. (4) While graphical methods
are available for the determination of this value (5),
Higgins and Siegel (6) have shown the advantages of the
method of complex-variable differentiation for this purpose.
24
The phase margin, being the angle by which the phase
angle of the open loop function falls short of 180 degrees
at- the gain-cross-over-frequency, and the gain margin, being
the value in db by which the gain at the phase-cross-over
frequency is less than z_ero db, are also measures of the
degree of damping of the· closed loop systemo H;igher values
of phase and gain margins indicate a higher degree of
dampingo A simple mathematical relationship between
phase margin and damping ratio has been recognised in
a second order system leading to the following approxi
mation for conditions where phase margin is less than
40 degrees: (4)
Since th~ step function represents the severest
kind of disturbance that a system might normally come
- upon, it has been c9nventional to study and stipulate
system performance regarding its stability and damping
in term~ of it_s transient response to a step inputo Since
the step function incorporates a wide range of frequencies,
it is reasonably expected that t'here may be considerable
correlation and correspondence between the ·nature of such
transient response and the nature of freqmmcy-~esponse
of the same systemo The existence of such inter-relation
ships between the frequency and trans;ent responses of a
system have ~en substantiated by several successful attempts
in developing methods of arriving at the closed !oop per-·
formance direotiy from the poles and zeros of the open
loop functiono
Biernson (1) has offered a combination of mathematical,
numerical and graphical procedures for this purposeo More
recently, Kan Chen (8) has described a simple method of
estimating the closed loop poles from the Bode plot of the
open loop functi.on. This method agrees well with Biernson'.s
earlier conclusions (9) that the closed loop poles of a
feedback control loop are "roughly• equal to:
•1. the open loop zeros under w0 in frequency,
2. the open loop poles over w0 in frequency,
3. plus pole roughly at -Woo"
Biernson (9) has also shown several useful rela
tionships connecting the frequency response and the tran
sient response to a step input. While the inverse rela
tionship between the rise time and the band width is
true for a first order system (5), he establishes the
general relationship:
26
and concludes that "the approximate inverse relationship
between the step response rise time and the gain-cross
over-frequency is very general and reliablena He also
br~ng~ out the correspondence between the peak overshoot
of transient and the value of ~a He also points out that,
while the mid-frequency poles determine the rise time and
overshoot, the low frequency poles -- equal to the low
frequency zeros of the open lo.op function -- determine
the nature of settlement of the transiento The contri
butions of the ~igh frequency poles decay too rapidly to
be perceptiblea
Biernson (10) has also offered a simple method for
calculating the time response of the closed loop to any
arbitrary input, by a process of splitting up the input
into several simpler components and super-imposing the
transients for the different componentsa The following
relationships are suggested for use in sketching the
transient responses:
la Rise time to a step response is rou~hly
equal to 1/wcga
2a Maximum error for unit ramp input is
roughly 1/wcgo
3o The maximum response to a unit impulse
is Wcga
Clynes (11) has developed "a simple ~alytical
method for linear feedback system dynamics" and offers
convenient relationships connecting the damping ratio,
the· undamped natural frequency of the closed loop. system
-- wn, and the error in terms of the zeros and poles of
the open loop function.
27
Perhaps, the most compr~hensive of the easier pro
cedures for obtaining the closed loop poles of the system
-- alternatively referred to as the roots of the character
istic equation -- is due to Evans (12)a The root-loci
are plotted on the compl~x plane and represent the complex
values of s for which the open loop transfer function
under variable gatn becomes equal to - 1. These originate
on the open loop poles and terminate on the open loop zeros
as the loop gain is increased from zero to infinity. The
sketch is easily made with the spirule. All the roots for
a given gain are obtainable. The values of the undamped
natural frequency of the system, the frequency of damped
oscillations and the damping ratio corresponding to the
least damped roots are directly obtained. The root-loci
are equally useful in analysis and synthesis. They are . .
particularly valuable for multi-loop systems. Reza {13)
has discussed usome mathematical properties of the root~-
loci"a
Many further studies have been made regarding the
conformal transformations between the frequency plane and
the s-plane. Jackson's (14) "Extension of the root-locus
method· to obtain closed loop frequency response", Chu' s
28
(15) "Correlation between frequency and transient responses",
Zaboroszky's (16) '-Integrated s-plane synthesis using a
z-way root-locus" and Chu's (l'l) ~ynthesis ••o• by phase
angle loci" belong to this category. Russel and Weaver (18)
offer a method of "synthesis of closed loop systems using
curvilinear squares to predict root-location•. Koenig (19)
discusses a "relative damping criterion for linear systems•,
which, applied to the characteristic equation of the system,
permits the evaluation of:
l. the sectorial regions of root-location
on the complex plane,
2. the root with the lowest damping ratio,
3. the root with the lowest frequency,
4. the number of roots with frequencies
greater than a given value.
Numakura and Mura OW) have proposed 11'a new stability
criterion for linear servomechanisms by ·a graphical method",
i.'in which the stable region of the system is drawn on a
plane with two of the parameters as coordinates"a
29
Very recently, Mitrovic (21) has presented a com
prehensive method of 'graphical analysis and synthesis of
feedback control systems', which treats the operational
transfer function algebraically and thus retains all
operations in the real domain.
The objective of the designer is to obtain equali
zation between the conflicting requirements of good stability
and low steady-state erroro In the design of the equalizing
system, an attempt is made to obtain the best possible form
of step-transient of the closed loop system, a process
referred to as optimization.
Several criteria for the determination of the optimum
form of transient response have been suggested. Schultz
and Rideout (22) have studied, with an electronic differen
tial analyser, the relative merits of different intergral
functions of error as criteria for optimization. These are
functions of the nature:
[ a: J F (eJ . rJ.f.
Analogue computer (23) techniques have been found
particularly useful for the process of optimization in
system design, since they permit easy variation of system
parameters and the rapid evaluation of its effect on the
transient responseo Fickenson artd Stout (24) point out
30
that, if- rectifier-thermo-couple voltmeter measures the
output of the closed loop analogue set-up and if the input
signal is a square wave, the minimization of the voltmeter
reading becomes an easy experimental method of optimization
on the basis of mean-square or mean-absolute error as the
criterion of optimizationo
Bo Analysis and Synthesis of" Non-linear Systems:
A non-linear system is one whose dynamics are repre
sented by a non~linear differential equationo The non
linearity may be due to certain imperfections in the electro
mechanical elements of the systemo It may also be purposely
introduced as a design requirement o.
The non-linearities introduced by imperfections in
apparatus generally take the form of a non-linear amplitude
responseo Common examples include dead space, backlash
and saturationo Electrical analogues for such non-linear
functions can be set upo (25) A versatile analogue computer, ··. ,·.
incorporating non-linear function generators, becomes-a
particularly useful tool in the analysis and design of ·
non-~inear systemso Abou-Hussein (26) has suggested some . ....... .·
interesting methods of representing non-linearities.with
linear elementso A resistive potentiometer shunted by
a proper resistance may also be employed to represent
31
certain continuously variable non-linearities in amplitude
response. (3) Hamer (27) has provided a mathematical basis
for 'optimum linear function generation' employing relay
or diode circuits.
The major difficulty in dealing with non-linear
systems is that the frequency response op the Laplace
transform methods are not applicable, since the principle
of superposition becomes invalid for cases of non-linear
amplitude response.
Several techniques of approximation have been at
tempted with a view to apply the frequency-response approach
to non-linear systems. The 'describing function' method
developed by Kochenburger (28,29), Johnson (30) and others
gives a close approximation to actual results. The
describing function of a given non-linearity is 'defined
as the amplitude and phase angle of the fundamental com
ponent of the response relative to the assumed driving
sinusoid' (30). This method has found successful appli
cation to systems with different kinds of non-linearitieso
(31,32)
Ogata (33) offers •an analytical method for finding
the closed loop frequency response of non-linear systems'
which takes into account 'the effects of all or most of
the ~igher harmonics of the non-sinusoidal output of the
non-linear element 9 o Prince (34) d~soribes·a 'generalized
methodV for the same purpos~ by the 'equivalent gain'
method, where the 9equivalent gain', equal to 'th~ complex
ratio of the.maximum value of the output to the maximum
value of the input', plays the role of the describing
functiono
An alternative approach ·for t~e analysis of non
linear syste~s is to plot the system trajectories on the
phase-planeo Kalman (35) has developed the tedhnique .of
decomposing 'the phase plane into linear regions within
which the trajectories converge to corresponding critical
poin_ts~' Kazda (36) has applied the phase-plane approach
to an analysis of the errors im:non-linear systemso
Non-linearities in servo-components sometimes provide
an advantage in the process of optimizationo A saturating
servo-system can be ·provided with linear compensation so
as to take advantage of the limiting characteristic for
better optimizati~no (37)
Recent trends in optimization .prefer the introduction
of carefully designed non-linearities _in the system.(38), .· ··.
in order to satisfy the increasing needs for improved
performanc_e standards o . The requirements of such.' adaptive
servo-mechanisms? (39) are accomplished by several methodso
The more common forms -··.appear to be: ~xtended switching in
a saturating system (40); derivative and integral controls
33
(41); piece-wise linear servos monitored by- an anticipator
unit (42); non-linear feedbacks involving multiplication·
of variables (43); designs with delay-type compensators
with the nec~ssary nature of frequency and amplitude
response {44); variably damped servos with damping inversely
proportional to error (45); computer-controlled relay
servo-mechanisms; etc.
Many mathematical and experimental studies of the
problems of the. design and analysis of such non-linea_r
systems have been attempted. The phase-plane approach
appears to be more popular.
The field of non-linear servo-mechanisms is pres
ently receiving considerable attention and making rapid
progresso It offers ample scope for development and
promises substantial improvements in performance standards
over the linear systems, where the extra cost would be
justified.
34
III. CHOICE OF BASIC SYSTEM
A simple type-1 third order position control system
is chosen as the basis for the study.
A. The Simple Position Control System:
A schematic diagram of a common form of a simple
position control system is shown in Fig. 1. The reference
and the control positions are converted to equivalent
electrical signals by the potentiometers with transfer
constants of Kp volts per radian. The difference between
these two is amplified and employed for correction. The
'difference amplifier' is an electronic one. It permits
the subtraction of the output signal from the input signal.
It also provides an amplification, Kd•
The 'amplifier' provides the increase in power level
necessary for operating the servo-motor. This generally
takes the form of a magnetic amplifier or an amplidyne.
In the latter case, the transfer function of the amplidyne
is of the form:
(1)
At"1PLIFIEII. MOTOR
tl t -E -4
Fig. 1: Common Form of a Simple Position Control System
+E
OUTPUT
f'oTEN,,oMET~R
35
36
The servo-motor is generally designed to have a
family of straight lines as its torque-speed characteristics
for different values of the voltage across its armature.
Under an inertia load, the transfer function of such a
motor is of the form:
~ km =k G =---
m ,,,, f {!+ 7;,J)
kM/fw,
f [ju~ ---- (2)
where,~ and Tm are constants determined by the electrical
and mechanical parameters of the motor and the system under
control and Km'• Km/Tm•
Under conditions in which the motor, in addition to
the inertia load, has also an external torque, L, applied
at the point of control, the following transfer relationship
exists:
f) == c.
where, 1n = L JeL
I<,.,, E(A. + Tm L'h,
j, { 1-r 7':n ;,) ---- (3)
represents the normalized load torque',
being the torque per unit effective polar moment of inertia
of motor and load with reference to the output shaft. It
is seen that the 'normalized load torque' has the dimensions
of angular acceleration.
The above transfer equations relating to such a servo
motor are derived in Appendix. I.
B. The Block Diagrams:
The block diagram of the basic system with unity
feedback is shown in Fig. 2.
37
Figures J(a) and J(b) show two alternate methods
of modifJing the above block diagram so as to bring out
the location and the effect of external load disturbances
on the system.
It is easily appreciated that the diagram of Fig.
3(b) is adapted easier for analogue computation tech-
niques.
c. Selection of a Representative System:
A value of l is chosen as the time constant of a
typical amplidyne: T = 1.0. a
A value of 0.1 for the time constant and a value
of 0.1 for the gain constant are obtained for a typical
servo-motor: Tm: O.l; Km: Ool. The values of the system
parameters which yield the above values for Tm and Km
are given in Appendix II.
38
'D1FFEIUNC/i AMPUF/fR MOTOR.
PoT£Nno-AMf'Lf t=t!R
METliR k(). kwe.. :r:NPUT
krA. OUTPUT
k, I+'!;)' j,(i+ ~p)
Fig. 2: · Block Diagram of Basic Position· Control System
kp
Fig. 3 (a):
L,._
Block Diagram of Basic System Showing the Location of Load Disturbances. (in terms of Poles)
39
Fig. J(b): Block Diagram of Basic System Showing the Location of Load Disturbances. (in terms of Time-Constants)
41
The forward transfer function of the representative
system chosen thus becomes:
/<I'. /<". k°" · k'W1 k O - ~~-=-~---------:- J{l-r 7',;../') (Ir 7;.,,f) f ( /+j,) {1+ O·tf)
= ko' ~OJ :;: (kf f<,1 ko. km)/(To. '?;,) : -*--=-~---- J,(f,+ 1:.) r ,.,. r..,; ;, o~,> (!,t,oJ
(4)
D. Characteristics of the basic system:
1. Frequency-Response and Phase-angle Characteristics:
The open-loop frequency-response and frequency
phase angle plots of this basic system are shown in Fig. 4.
A sketch of the root-locus of this system is s.hown in Fig. 5.
A programme on the '24.2 System' of the campus
digital computer for obtaining the necessary data for
plotting the frequency-response, the frequency-phase angle
and the Nyquist plots of a type-1 open-loop function with
four or a lesser number of zeros and with seven or a lesser
number of poles is given in Appendix III. An extension of
the programme is also given for the simultaneous evaluation
of the closed-loop frequency-response of the same systemQ
This extension is applicable only in such cases where the
feedback transfer function is unity.
42
I\) ~ (JIO)-.J(D(OQ
I I ~ ~
r + r
t rt1t11tttttt
L I I I
I
~ I ,
, r fl
IIH/ 11111 11
l I I
~I I
I ! !
j I
'I 1 ~ !
' 11 I
=
43
le I
j, ( jo+t) {j:>-tt(?)
(0
• .,
' 6
-10 _., -· -7 -6 -5 -+ -3 -a.
Fig. 5 :Root Locus of Basic 'System
44
Appendix IVo provides a similar programme for obtain
ing the data for plotting the root-locus of a system with
a similar open-loop function.
From these plots it is observed that the phase
angle of the basic S¥stem becomes 180 degrees at the phase
cross-over frequency given by:
---- { 5)
where, P1 = 1/Ta and p2 = 1/Tm are the two non-zero poles
of the basic third order type-1 system. The value of
system gain constant, K0
, necessary for obtaining gain
cross-over at the same frequency is found to be approx
imat el:· 10.
2. Steady-state Errors:
Since the system is of type-1, the steady-state
error under step-position input is zero.
The velocity error constant of the system is given by:
---- (6)
The steady-state position error of a type-1 system under a
constant velocity input of velocity Vis given by:
---- (7)
Hence, the ultimate design ought to have a velocity error
constant of at least 30~ for the error to be less than 1°
umde r a steady-state velocity input of 5 rpm.
45
The system has unlimited steady-state position error
for a constant acceleration input. But, such an input is
unlikely to occur in such systems.
E. Degree of Compensation Necessary:
It is easily seen that the gain-cross-over frequency
of the over-all system has to be close to the phase-cross
over frequency of the basic system. A higher value renders
the design of phase advancement difficult. A lower value
makes an unnecessary sacrifice with respect to the rise-time
of closed-loop transients. But, if the over-all gain
cross-over frequency is chosen to be close to the phase
cross-over frequency of the uncompensated system, a single
lead network is sufficient to obtain a phase-margin of
40-45 degrees and thus ensure reasonable degree of damping.
Since the low frequency asymptote of the frequency
response of the basic system with gain-cross-over close to
phase-cross-over crosses the zero db axis at 10 r/s, we get
a value of 10 for the Kv of the uncompensated system •. The
introduction of the phase-advancing (lead) network under a
restriction of the gain-cross-over frequency range is expected
to reduce the gain at the operating frequencies by .two to
four times. Hence, in order to obtain a low frequency gain
constant of value above 30, a gain-advancing {lag) network
of a gain of about 20 db becomes necessary.
IV o DESIGN OF PHASE-ADVANCEMENT
A single lead network providing about 45 - 50 degrees
of phase-advancement at a gain-cross-over frequency of
about 3.1 r/s is found to be suitable for the purposeo
Ao The Lead Network:
Figure 6 shows a common form of lead networko With
a sinusoidal input voltage, the voltage across the .resistance
R2 leads the input voltage in phaseo With zero source
impedance and infinite load impedance, the operational
transfer function of the lead network is given by:
Ea . E.
/)1,
::: ().. .
where T = R1c1 and a= R2/R1 f R2• Usual values for
(R1 f R2)/ R2 = 1/a lie between 6 and lOo
(1)
It is observed from equation 1 that the lead network,
while provid~ng the phase-advancement, causes a reduction
in the system gain-constant by a factor a. The lead network
introduces in the system a zero equal in value to its
lower corner frequency of 1/T and a pole equal in value
to its upper corner frequency of 1/aT. Hence, it provides
a low-frequency attenuation of a factor: _L / _L: ao T aT
47
R,
c,
Fig. 6: A Simpl~ Form of Lead Network
48
An analogue set-up that simulates such a lead network
is shown in Fig. 7.
B. Cross-over Frequency and the Lead Network:
In Fig. 8, the zero and the pole of the lead network
are plotted on the negative real axis of the p plane. The
phase-advancement provided by this pole-zero configuration
for a sinusoidal input of frequency w is shown by the angle
¢, as marked.
The phase-advancement at a given frequency w is
given by:
---- (2)
For obtaining maximum phase-advancement with a given
ratio, a, we equate the derivative with respect to T to
zero:
= (3)
This equation simplifies to the condition for maximum
phase-advancement as:
2..
TfA:Ji.a=l,I I
T ---- ( 4)
/. 0
J.o
=
/. 0
a< l.o
(!+ Tj) . a.. (I+ a 1/')
49
Fig. 7:. Analogue Simulation of a Simple Lead Network
I'=_, a.r z = .J_ 7
Figo · B: · Pha:se--Alivanc-ement· with a Le~d N·etwork
50
51
Thus, the lead network provides maximum phase-advance
ment at a frequency whose value is the geometric mean of
the upper and lower corner frequencies of the network. 1
Since (zero)/w = w/(pole), - a~, we arrive at the logarithmic
relationship: log z - log w = log w - log p.
Hence, for maximum advantage in phase-margin from a lead
network of a given ratio, 1/a, the upper and lower corner
frequencies of the network should be equidistant from the
final cross-over frequency on the logarithmic scale of
frequency.
If the final gain-cross-over frequency is thus made
equal to the geometric mean of the corner frequencies of
the lead network, a s~mple expression gives the increase
in phase-margin provided by the network in terms of the
ratio of the network.
Substituting the relationship w = 1/T.a! into the
phase-advancement equation, i.e., equation 2, gives:
¢ -1 Ii -1 ! : tan 1 a - tan a, max.
which, under a trigonemetric simplification, becomes:
-1 ! . ¢maxo = 2tan 1/a - 90deg. ---- ( 5)
Figure 9 gives a graphical plot of the maximum
phase-advancement of the lead network against the ratio
of the network, 1/a.
52
r, = L t().n' ( /;t) - 'lo~
I ~ + 6 10 ,.a ,.,. ,,
LEAD NETWO~K RAr,o ( ;J--.
Fig. 9: Maximum Phase-Advancement from a Lead Network.
53
The final value of phase-margin of the phase-com
pensated system becomes, under a maximally designed phase-
compensation,
~ _ ,2, A.,,, -I _L _ /;;..' (,.;CJ _ ~I f4_ f. . r;,,, (~(A"/() - ~ ); f>.t, ---- {6)
c. The Basic System and the Cross-over Freguency:
The gain-cross-over frequency has thus an important
role in the maximization of phase-advancement with a lead
network. A knowledge of the effect or dependence of the
cross-over frequency on the parameters of the basic system
is necessary for estimating the over-all performance of the
compensated system.
1. System Gain:
The basic third-order type-1 system is of the form:
£, I I
I\ (Yj = 0 "
io1
The phase-cross-over frequency of such a system is given
by wcp = (p1p2)i. The gain-cross-over frequency wcg may be written in terms
of the phase-cross-over frequency as:
~; - 11~r -- //}J;. ---- (7)
where, A ~/< A (.a.I A < ~-l< A - ,I
/;,; ~cJ CJo/1 P-t
54
The system gain necessary for obtaining gain-cross
over of basic system at the above value is given by: ;.,.7¥,. r z 1,~7 h
~ I ~ [ /JfJJ. [ l'~f.i +J'_j . ul)/tfi + 41 · This equation simplifies to:
fn 7 %. ~ ' = /); A · [ ( ~;, + f J (1' A -t- /}; . ---- (8)
When wcg becomes equal to wcp' the above reduces to the
form:
---- {9)
Figure 10 shows a logarithmic plot of system gain
against frequency, as represented by the above equation.
It is seen that the logarithmic plot is a straight line
in the regiOn of values for wcg from ); . [ IVJ,,]t to ~ ! tJ.J:A!f;J +. The value at wcg = (p1p2) is known to be
p1p2 (p1 f p2 ). The value of the function at Wcg equal to
p2 becomes: KI ==- I~ · .Ufa. I), 2--r t/ .
{1= I hfl,) while the value at wcg equal to p1 is:
): ~ = J/',11,_. ) = J1 Jz fl" /Ji L -(- r: • Hence, the slope of the logarithmic straight line is given
by:
'11. -", /, (,,, = µ;;;;) - 41; f (/. ffi/A)
,4,A - 4'J)1
9~ 8 7
6
s
4
3
2
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:;;? - ' -i- . -+- - - -'- + .. ~- . ;:=-i==; ,-+--+--+-+--1--f
---+---l--l----1---,·--+---+--,-- l ----, --+--l·--~-+ -1--+--Je--t---L---l----,--+--+--·-+----,-+--~---- ,--,----1-=r-=£=F_ --~ --+----- ::::t-: - -+ ----- ! - --+--+--l--l-----+-- ; - 1·- --+ - ·-+-i-- l--t--l--'--'---'---- l--- -l-- l---l- ---l----4---'-- ·,--+---~- _, ___ _, +-·-·t-f f- ( - -,--+-l-- ,--- -+ ·--,-+-- -'-·--+-·- ·
i _____ __ ,_ ' : , ' : - ~- , +·r -+- L-,--,- ' : 1
~---+--+---+--- +---~s- +--~- 1---+---• ---~--+- t--f~i---+-+--+-•--+--,; -- ,-+ T ; • -~~--r~ !--:'-· -~ -- '- -' • - 1--+--+-•
I ' I I , • : i , i I I
55
which is easily simplified to:
4!!0l 4; [Alt,}
56
---- (10)
A graphical interpretation of the logarithmic linearity
between gain and cross-over frequency yields a convenient
relationship between the gain-cross-over frequency and the
gain of the basic system:
/ ' ,lo i' -,..1,[~tv~-~;~ M/ ,tf.07) = °J (.if~ ,.lf:t.,)
= ~-/K(i! 'jj;j;) . ::iJ ---- (11)
Substituting the value of gain at wcg: (p1p2)i as given
by equation 9, we simplify this relationship considerably:
K (c.;. J = w/. C;,, + h) r !i y. ~ ~ % ), [IV;] ~ < w CJ < ,A jl'~rJ '.
---- {12)
where,
The departure of the gain-cross-over frequency beyond the
above limits is rather unlikely. Even in such cases, a
quick estimate of system gain is possible on the basis of
Fig. 10, where it is seen that the correct value becomes
50% greater than that given by linear approximation at
the extremities of wcg: Pi and wcg: P2•
2o The Gain-constant of the system::
If the system gain of the basic system is obtained
as above for a particular gain-cross-ov-er frequency, the
corresponding value of the gain-constant becomes:
t~,) = K~e;;jll~ J. The Phase-Margin:
A semilogarithmic plot of the variation of the
phase-margin of the basic system against frequency of
gain-cross-over is shown in Fig. 11, for frequencies
ranging logarithmically from p1 to p2•
Do Lead Network and System Gain:
57
(13)
On the basis that the gain-cross-over frequency of
the system has to be kept the same before and after phase
compensation by necessary changes in system gain, let the
value of this frequency be wcg• It is logical that, since
the lead network incorporates a certain degree of attenuation,
an increase in system gain may be necessary for gain-cross
over to be unaltered. The ratio of the over-all gain for
the system with the lead network to that of the basic
system is given by:
sf ~ -(- ( f,t£J/ h. i { l'~t -C..-t_~J.12 t{' ,
.,_ r~l k ( tc," Oo,t,./'•"'-'"''"--!) "- J.
~
Since the lead network corner frequencies are chosen for
maximum phase advancement at the frequency of gain-
Fig. ii: Phase Margin of Uncompensated Thi rd Ordo.l4 'r,rpe-1 System for Different Values of Gain-Cross-Over Frequency
59 ·
cross-over, this equation simplifies .. as below:
I<' I . (14) -· /(_ I
. o. ,.
Henc.e., · when· a lead _network of ratio . of lower to upper corner
frequeric'ies a. is intr.oduced., the gain of ~he original system
has to be increased by ii: ratio (1/a~i·fo order· to inai~tain ,· .·•, ...... ,·
·the gain-cross-ove~ .. _frequency unaltered arid to obtain
maxii;nization of. phas.e-margin·o
Eo Lead Network and Gain Constant: ·•
The ~.Ver-all, ~ransf er function of the phase~com.-. .
pensated system b~cQme.s: .
Ko 1 · {!,+ ;/:)
7:: . f {j, + },) {t6 + JJ { l.,. f,) ---- {15)
._The ··g~~n-'.'""constant or the velocity-error-coeff~cient · of
. the -system is then given ~y:
. *r,145 ... ~~~s""-f":). .
-. · ·k~ '-· . . Yr ·· --· . : . J;;,: .' ·ft.h. _!..;.··. . . . . (}. r
(16)._
Thus, when a le~g ,_·net~prk of ratio of lower to upper corner
·.frequency a .is· introduced; along with gain adjustme!,lt to
maintain the gain-cross-over frequency unaltered, the
velocity-error-coefficient is reduced to a value (a}!
times that of the uncompensated system.
60
F. Generalized Procedure.for Phase-compensation of Third
order Type-1 Systems:
The information contained in Figures 9 and 11 are
.combined into a chart in Fig. 12. It gives the values of
resultant phase-margin of any third order type-1 system
when it is maximally compensated for any cross-over frequency
and with a lead network of any ratio. These values are
marked on a logarithmic plane with the cross-over-frequency
along the X-axis and the lead network ratio along the
Y-axis. On this chart are drawn the loci of points where
the _phase margin. of the compensated system becomes 30,
40, 50 and 60 degrees. The region bounded by these loci
represents t~e region where the possible choice of maximally
designed lead network lies.
The designer normally has the choice of phase-margin
necessary depending on the degree of damping required and
on the relative susceptibility of the system to oscillate
as evidenced by the relative magnit~de of the maximum
value of closed loop frequency response. This is, in
general, a decision left to the designer's judgment to
61
1' lo~ ,, ,l.o
4
t ":'""\ 'I~ '-.i
0 ......... ,.....
~ L
~ o ----ullMll,l1
~ 0 ~ .... ~ ~ ~ '1
1'1 0 -l'I -41 /',-( ~ )'4 ,, !AA l'r(f:)34 P.i
Lo/J fAJcJ --.. 1" = t.1(~)'~
Figo 12: Phase-Margin of Phase-Compensated Third-Order
Type-1 System £or Different Values of Lead Network Ratio
and Gain-Cross-Over ·Frequencyo
62
be based on his experience. Allowance has also to be made
in choosing phase-margin for the reduction that may be
brought about by the introduction of a lag network.
A given phase-margin can be obtained by several
designs employing different ratios of lead network and
corresponding values of cross-over frequency, as repre
sented by the phase-margin locus on Fig. 12. But each
different choice implies differences in system performance
and requirements concerning:
(i). system gain, which increases with cross
over frequency and decreases with lead network ratio,
(ii). system gain-constant which increases
with cross-over frequency, but decreases with lead network
ratio, and
(iii). the rise time of closed-loop step
transient-response which decreases inversely with gain-
cross-over frequency.
A particular point on the locus of the required
value of phase-margin has to be chosen taking into con
sideration all the three criteria mentioned above. The
system gain is estimated as:
= ~~ (f +fJ.)._L. , ~ (17)
63
The system gain-constant is calculated as: ·
---- :{l~)
These values are always iess than the correct valueso
The error is ~ero when w~g: wcpo It is within 5% if:
}.,. ( ,,.) ~ . tJ.J. . b (. )., ) % fl }, ..:::::: ~ ~ ;1 ft .
The rise time of closed-loop step-transient-response is
approximated as:
. (. -~
+ /O ~.
(19)
(2~)
A few trial choices may be made, if necessary, and, in
each case the gain, the error constant and rise time esti
matedo · The best choice could then be decided upon, deter
mined by the desig~er's discretiono
Go The Choice of Lead Network:
As a compromise between rise-time of transient and
error-coefficient, the value of the cross-over frequency
is taken as equal to the phase-cross-over frequency of
{p1 op2)i,_ equal. numerically to (lO)lo A value of final
.. phase-margin of 50 .degrees is tentatively chosen for the
64
systemo · Allowing a reduction of 5 dego due· to lag network,
the lead network has to be designed for a phase-margin or 55 dego The locus for 55 dego in Figo 12: crosses the cross
ov~r ·r~equen~y of 101 at a lead network ratio, 1ia, of lOo
Hence, the corner frequencies of the lead network for
maximiza~ion of phase-margin are:
~,~ c/io. I =- I = -V/0 )
and 1v,.-L ::: ,m,. Jio = /o. (by eqno tfi.
The gain required for the compensated system becomes,
with zer~ error,
4)
/(,'"' t.Ji {!, +fa) :h · = Pio//!. Ji"o = 347. (by eqn. 17)
The corresponding value of the gain-constant of the com
pensated system is:
le = k = ~ '-{f,+ ~)· €. = _!. · -1-. t,{lioJ~ 3-47. ll ~ r 1
I~ j,JJ.,, Jio /./o (by eqn 0 18)
Thus, the transfer function of the phase-compensated system
becomes:
KG - (phase compensated)
--f (}+1) (j+io) C/+:'o)
.:34-7 ·. -
I {J+1ev2.
3. 4,7 -----------.t. f ( I+ (J.tf) -.--... (21)
V. THE DESIGN OF GAIN-ADVANCEMENT
A simple lag network designed to provide a low
freqaency gain-advancement of 20 db is found to·be
suitable.
A. The Lag Network:
65
Figure 13 shows a common form of lag network. With
a sinusoidal input !oltage, the output voltage lags behind
the input voltage in phase. With zero source impedance
and infinite load impedance, the operational transfer
function of the lag network is given by:
.:::: ..L . ( ,J, + -;foJ a- {f+~)
...
(i+ rfa) ---- (1)
a, determines the low-frequency gain-advancement obtained.
When the system gain is increased by a factor a,
the function become~:
(; +#) (!+ ~)
- ().. (t+~f}
{I+ ~'tf)
Hence, an increase in the error constant is obtained.
(2)
An analogue set-up that simulates equation 2 is
shown in Figo 14.
R.,
c,
Fig. 13: A Simple Form of Lag -Network
E1i ·.
/. 0
(.0
=
,. 0
(). .> /. 0
{I+ r/l) a.. ___ _ (I+ a'tf')
Jc'J..g., 14: -. Anai;ogu~ Simul.ation -of : a Sims).'?. ·aai1?--'Advancing Syst_em
67
B. Low-frequency Gain and Steady-State Error:
In order to restrict the steady-state position
error under a maximum constant velocity of 5 rpm, the
minimum value of gain-constant required is found to
68
be JO. (from eqn. III?). The gain-constant of' the
phase-compensated system is found to be J.47. Hence,
the gain advancement is required by a ratio JO/J.47.
Gain advancement by a factor of 10 and a lag network
providing a high-frequency attenuation of 20 db is found
to more than satisfy the requirements.
c. The Lag Network and System Optimization: ·
It is observed from equation 2 that the gain
advancing network introduces into the system a zero
equal in value to its upper corner frequency, 1/T, and
a pole equal in value to its lower corner frequency,
1/aT. t!f Tis large, both zero and pole are near the
origin and show no stabilizing effect except to intro
duce another root which is near the zero (located at 1/T)
for all values of gain. Such a root gives rise to a
response with a long time constant {approximately equal
to T). As Tis decreased, the zero approaches the other
roots and has a stabilizing effect. For still smaller T,
the pole becomes effective and exerts a destabilizing
e£fect'4. Thus the values of the corner frequencies of
69
the lag network, relative to the values of the corner
frequencies of the rest of th~ function, largely determine
the performance of the. over-all system.
An optimum system may be defined as one having the
best form of step-transient response. The best form of
step-transient response may be considered as the one
that has the least value of time-integrated square of
error. An experimental study, by analogue computer
techniques, is made with a view to determine the values . .
of the corner frequencies of the lag network for which
the system gives optimum response ~o step inputs. The
effect of the lag network on the nature of load-dis-
turbed transients is also examined.
D. Experimental Procedure:
An analogue computer set-up simulating completely
the basic system, the phase-advancing system and the.
gain~advancing system is shown in Fig. 15.
The simulation of the basic system follows the
block diagram of Fig. 3 (6), where the_ point of application
of load-disturbance is brought out. The step function
representin~ load-disturbance is applied at this point.
For the ·system chosen, Tm= 0.1 and Km= 0.1. Hence,
o., /.o o.t
0,6"
INPllr o., IO /.o
0.1
l·O (!+ ()./P) c {l+O.fil!t}.
LoAD 1J,s TIJRSANCE
OUTfUT
/0 o./
(/ + O.f J,)
.3 3. 5 {Ir/,){!+ t:J. 5 C j:,J
,J, ( 1+ ;>){,-,. 0.11')(1+,uf,J {t+ 5 cp)
Fig. 15: Analogue Set-up of Compensated System with Variable Design of Lag Network.
Km/Tm: 1. Also, the polar moment of inertia of the
system is loO slug-ft~ Hence,
normalized load torque = (actual load to.rque)/
(polar moment 0£ inertia 0£ system)
- actual load torque - (numerically).
The simulation of the phase-advancing system
follows Fig. 7.
71
The simulation of the gain-advancing system
follows Fig. 14. If the value of the two equal-valued
capacitors, C, in Fig. 15 is made variable simultaneously,
we obtain a variable function for the gain-advancement
system:
KG = {Gain-advancer)
10 { 1 +- o.5c))
{!+ 6 cjo)
(J _,. t1 ---- (3)
Thus, by changing the value o:f the capacitors, the
position of the lag network on the frequency scale can
be changed over a wide range. The leakage resistances
across the capacitors are measured and allowed for in
the set~up 0£ the analogueo
The equalizing systems are placed between the two
parts of the basic system simulating the amplidyne and
the servo-motor.
72
For different values of the corner frequencies
of lag network as obtained for different values for the
capacitors, a low-frequency square wave (0.01 c/s) signal
is applied- at the input and the nature of output variation
is observed and recorded on a 'Brush' recorder, when
the input signal is suddenly removed.
For the same amplitude of square wave signal
applied to the load point, the behaviour of output
signal is observed and recorded, when the load signal
is suddenly removed.
E. Evaluation of Results:
Several interesting and important details are
borne out by the results of the experiments. The results
of the experiments are detailed in Table I. A repre
sentative set of recorded transients under input- and
load-disturbances are shown in Fig. 16. Figure 17
shows the frequency-response and phase-angle characteris
tics of the system for the different values of the
capacitors. Figure 18 shows the Nyquist plots of the
same functionso In Fig. 19, the root-loci of these
functions are shown. The locus of constant gain is
superimposed on the root-loci. Table II summarizes a
few of the results obtained from these graphical plots
along with relevant experimental results.
73 ·TABLE I
Lag Network and Transient Performance.
Transfer Function uem.er r'.1.-.. .. - Nature of Input-Step-Transients. Nature or Load-d1sturbed No. Value of of Gain-Advancing hf L::.v Net work Trinsients
C in uf Rise Time Settling ~. output Time Constant System Lower Upper t Overshoot Tr Time fD:lsturban c e of· decay
1. With put Gain-Advancing : ystem. 75 % o.8,sec 1.0 Sec 0.06 Very Little dee av
i lOfl ~ 0!22T) 0.4 4.0 Oscillatory 2. uf I 2.5p r/s r/s 80 % Oscil: .atory 0.05
3. 1 uf 10H ~ 1~i 0.2 2.0 56 % 1.0 Sec 2.8 Sec 0.055 Oscillatory r/s r/s
4. 2 uf 10H ~ t6i>1 0.1 1.0 29 % 1.0 Sec 1.8 Sec 0.06 1. Sec r/s r/s
5. 3 uf lOf 1 ~ 1. 2T) 0.067 o.67 1.2 Sec 1.8 Sec 0.06 1.5 Sec r/s r/s 16 % I l5p
6. 4 uf lOfl ~ 2g) 0.05 0.5 14 % 1.J Sec 2.0 Sec 0.06 2.0 Sec r/s r/s I 2 p)
7. 5 uf 1011 ~ 2.21l 0.04 0.4 10 % 1.3 Sec 2.1 Sec 0.06 2.5 Sec r/s r/s 1 25p .
8. 6 uf 10(1 /. 3p) o.'.Ps3 o-}; JO% 1.3 Sec 2.7 Sec 0.065 J.O Sec (l f JOp) r s r s
9. g uf 1011 ~ ~l 0.025 0.25 30 'fo 1.5 See J.2 Sec 0.065 4.0 Sec r/s r/s .1 1p)
10. 16 uf 10(1 ~ ~l 0.0125 0.125 37 % 1.8 Sec 5.7 Sec 0.065 8.0 Sec r/s r/s 1 8 p)
11. 26 uf 10 (1 /. l2p) 0.0083 0.083 40 % J.1 Sec 9.9 Sec 0.07 2.0 Sec .,
{l f 120p) r/s r/s
... _12..... ~-JO uf lOfl ~ 12gl ~.0067 0.067 40 % J.4 Sec 11.J Sec 0.08 15.0 Sec r/s r/s 1 !5 p)
• r- ·-··· ·
. I
• ·' • '
- ---- ~i,-''
. +-t= ~ --+~1--
I k>N O.F CLEVITE CORPORATION CLE'\ ......---; ... ,<1· '·:,X'.:. ·=,:-,,::,:==;J.'.=. ~- . ::::::It===:z::::;;:::_+:,( __ ;::::_7-J ~- -=1.t
:
-_ -----:t+~t:-::::::',.,-·- ··--- -~ ·-·---=+--~-',-
i f"F i
{cJ
.·
·--: ·---~ :· __ -·_:·· . -~::::;···:·,,··< •;'.'<~--,; ~"i;-~_~;:.-t·.~--~;·:,-::;:~--~-~--w-,•9.~~:7~
I '"~'!I.~~- ·,· ·0.01 VI ~h.. 'l'i,d ~ '(
·:::· __ ·:'W~? "' '.··\'. '.·,._,~·~·-i= .·.~s~"r:_~.:-.~~~~7·7"-7•·~'.~-~,"'.'.:(t~ .-·Im ml ~~i'.· : · 0.01 v /,1 · l, ·..;;
74
' '
. i
;
'
',
-
' .
'
Fig. 16: Input- and Load-Disturbed Transients Eor DiEEerent Positions _of. .. Lag Network on- the Frequency Scale. {Cont.) ,- -----
f ·.5" h.s~/$a~. ;' 0-1 V/cJ.... /.,.:iu. l r.~ '
IJ-. ~~ ·r'- -1 .
C = 5 !-4-F ;
{f)
' PRINTED ·1N U.S.A. + . - - - -~ ~-. · 2· -4.,u,<,. ~ -v· ~ .- .. .. : .... _ ct ~ ~:"'-'C~~.,.~,-~ --: .
7.5
r··---- · .. -··\ . -. -_ -·~ ~-----= --· "">{ I I >h.,.-/Ssc. ; o.o, V /e,J.. L11e.j -
o. 06 7 12/.s ~· h. = O· 006 7 Jz4 · °1 , .
. i fttt.(.;./ sec. ~: c~s:=;~;;·,;;~7;.tJlit:-.~~-::~ • !
~ I
-'-
! 32 · ___ BRUSH- INSTH.UMEN".J i..,. __ .. s .... ·:._<P" -... - ; ... ·, ·- -~ ~ ~ -~ _ _ ,.,.~_<::_ -:i-,.~,-~r-"2"::""6 ¥ ,...., ,, •. ., ~~ --':."'~i.,_." •... +~.? ... ~.#·~·-~----.J
Fig. 16: Input- and Load-Disturbed Transients for Different Positions of Lag Network on the Frequency Scale.
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II = ,.
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Figo 18: 'jolar Plot of Frequency Response and Pha~e-Angle of the Compensated System for Different Positions · of Lag Network on the Frequency Scaleo
Ii = /.
-6
F_igo 19:
-5 -4-
Root-Lo.ci of' the Compensated System for Different Positions of' Lag Network on the Frequency Scaleo
TABLE II 84
Lag· ~etwork and· -System· Charact·eristics · ..
Corne~ Freq. ~ No. c •. wcg wop Gm .. t ts
Lower Upper . .m r ,,
0 1. 0;002 100 1000 14.5 9.5 -26 - 9 db Nega- ....... ....
uf r/s r/s -tive
..'.:.~.} 0 6. db'· 2. 0.5 uf 0.4 4.0 4.3 5.8 6 0-.07 0.3 ....
r/s r/s · . , .. .. See .... ··:, . .. .. ! ..
o· 0.52. o., 1 Sec 3. 2o0 ur 0.1 1.0 3.2 9.5 ~9 ~.~.o. ;·
"r/s r/s· . . ' . d'1 Sec
48~, o.s 2.1
4. 5 oO, ur Oo04 0.4 . ) • :L! 9 •. , 15.5. Oo54 See Sec . r/s. ,.,
r/s db -~ -- 2.7
s. s.o ur 0.025 0.25 J~O 9.5 52° _15.5 o._5J5 _1.J. Sec r/s r/s db Seo
6. 0.0001 0.001. 3:.0 o·
16 .0.55 2000 uf 9.5 57 db .... -r/s r/s ~
85
Th~ effect of variation in lag network design ·oh
system performance may be summarized under the following
heads:
lo Stability and Damping:
The system without gai~-advancement shows.a high
degree ~f stability and dampingo But a large error.is to
be expected in the st.eady-stateoThis ·is borne out by the
very low tendency for corrective action when the system
output is. shifted due to .a load disturbanceo (~igo 17oaoiio)
For extremely small. values of C (Nool in Table II),
the system is oscillatory (Figo 18oao) This is because
the gain-cross-over frequency is shifted considerably into
the unstable region (Figo ~7oao)~
For small values of C (Noo2 in Table II), the . . .
system is still unstable (Figo 18ob}o This is because
the pole of the lag network is close enough to the rest
of the poles· and the zero and exerts a destabilizing effect,
on the otherwise stable systemo
As the value of C is increased furth~r, the pole
of the lag network is shifted to lower values and hence
its· destabilizing effect is redueedo Thus the system
becomes more stable and better damped, as evidenced by
the changes in overshoot, {Table I) phase-margin and·
damping·ratio. (Table II)o An optimum value of C exists
86
with reference to stability and dampingo A maximum value
for damping ratio of least damped roots is obtained at
this condition (Table II. No~4.). This condition is .. ,. ..
· approximately represented by the point at which the
tangent from th~ ~rigin touches the constant-gain
locus on the root-loci (Fig. 19). This condition, in
this ca·se, is found to correspond closely to C : 5 uf ,
and to 0.04 and 0.4 r/s . as the corner frequencies of the
lag network.
As the value of C is further _increas~d, stability
becomes poorer, t~e overshoot becomes larger and the
sy~tem becomes more and more sluggish in responseo The
decrease in stability may be traced to the tendency for
the formation of' a double ·pole at the origin. The -
sluggishness is due to the fact that the low-frequency
zero of the lag network gives ·rise to a negative real
root of approximately its value. As C is increased,
the time constant corresponding to this root becomes
larger ahd the system ·becomes slow in response.
For very high ~alues of c, the pole :and zero of
the lag network appear to cancel each other at the origin
and the sy~tem. behaviour approximates that of' the system
without gain-advancement. The root-loci plots clearly
87
bring out the tendency .for the system performance to
approximate that of the uncompensated system as the value
of C is increasedo
2o Rise-time o.f Transients:
Table I shows a steady increase in the rise-
time o.f input-step transients as the value of C is
increasedo Table· -II brings out a corresponding decrease
in the gain-cross-over frequencyo While the value .of
rise time is found to change in the reverse direction ·as
the bandwidth, it is also noticed that an exact inverse
proportionality do.es not ;exist between these two para
meters under non-uni.form conditions of damping and
stabili t): o .
3o Nature of Settlement of Transients:
A steady increase in the- settling time of input
step-transients as the value of C is increased indicates
the increasing sluggishness o.f the systemo This is also
due to the -increasing time-constant of the negative
real root introduced by the zero of the lag networko
4o Transients under Load-Disturbances:
The absence o.f the lag network leads to l~rge
errors in the steady-stateo A load disturbance under
this oondit·ion is found to cat1se a large dislocatioD:
with very little corrective act'iono (Table Io Noolo)o
The nature of the lag network.does· not have any
appreciable effect on· the maximum shift of·output·under
a given magnitude of step-load-disturbance, except for
a small increase as the value of C is increasedo
The importance of the.zero of the lag network as
the determinant of system_settlement characteristics
gg
is clearly seen in the study of load-disturbed transientso
The decay of output under a step~load-disturbance is Cound
to be exponentialo The time-constant of this exponential . .
decay is £.ound to be equal to the time-constant repre- .
sented by the uppe_r corner freque~cy {the zero) of the
lag networko
. .
Fo Restrictions on the Choice of·the Lag. Network:
Sinc·e the upper-corner· frequency· o:f the lag
network is found to l~rgely determine the nature of
settlement of transieµts, it is eas~ly appre~iated that
the designer may ·attempt a prediction of a lower limit
on the zero of the lag network on the basis of the ·
settlement requirementso Especially in.the case 0£
load-disturbed transients, where the decay ·of transient
is practically exponential, a simple correlation between
the settling time and . ti.me-constant is ·possibleo It- is·
observed. (Table I) that, under conditions of good damping,
89
t-he settling time under step input is found to be approxi
mately equal to the time-constant of load-disturbed decay,
which, in turn~ i~ equal to the reciprocal of the. upper
corner frequency of th~ lag ·network. This leads to an
empirical and approximate relationship·:
T - t - 1/z max - s - min' (4)
where T. is the maximum value 0£ the time -constant max corresponding to the zero of lag network, t
8 is maximum
allowed value for the settling time of the system under
input or ·load disturbances, and zmin is the minimum
value £or the upper corner frequency of the .lag network.
Such a relati~nship is approximately true only when the
overall. system is designed fo-r a high degree of damping.
But, it may still be a use[ul guide in initially fixing
. the upper corner frequency ·or· the lag .network, so that
proper allowance may ~e made in . the design o1 phase
advancement for the required degree 0£ damping. The
lower corner frequency is then fixed on the basis of
the gain advancement required.
G. Choice of the Lag Network:
The choice of the lag network is £ound to be
limited to a small range of values for c, between 3
and 5 uf', in order to have good damping . and stability
and reasonably low values of rise-time and settling
time. The value C = 5 uf is chosen on the basis of
minimum overshoot in the region.
90
An examination of the frequency-ree,onse charac
teristic of the system with C equal to 5 uf suggests
the existence of a simple, though approximate and empir
ical, relationship connecting the optimum region for
the zero of the lag network with the zeros and poles
in the vincinity of gain-cross-over frequency. For
this case, we observe:
where
log w6g - log zlag = 2 log p 2 ---- (5)
w6g is the asymptotic cross-over frequency and
is the zero of the lag network.
An extended interpretation of this relationship
is found possible. In Figo 20 are shown on a logarithmic
scale the asymptotic cross-over .frequency and the zeros
and poles its vicinity. Taking into consideration the
approximate linearity of semilogarithmic plot, in this
region, of the frequency-response of the best-compensated
system, the following empirical and approximate rela-
tionship may be written: ---- {6)
~~l - -&; I 1',; ,~ T 4 A,-I
~; t.)GJ
t; ~, ,....., , Wcj
+ I', (,.)CJ Zt., z~
91
0 O I I
Fig~ 20: Typical Placement 0£ Poles and Zeros in a Compensated Third Order Type-1 Systemo
which simplifies to an expression for the best value
of the zero of the lag network:
(4/:;/) ---- (7)
In the case under study, the asymptotic cross-over
frequency being 3.33 r/s, the value is:
3 {3. 33:) · f.
·.,. :: 0.3&7. ---- (8) /. /0. /CJ
a value that is in agreeD)ellt with the experimental ·--
results of 0.4 ~/s ·for the upper corn~r frequency for
the best choice of lag network.
It is expected that the above form of expression
may yield a rough estimat.e .for suitable positioning of
lag network for higher order systems also, provided
all the zeros and poles in the vieinity of gain-cross-
over are taken into consideration.
With the value of 5 u.f chosen as the best value
for C, the transfer .function of compensated system
becomes:
92
33.3 (1+j) (1+.J.5j,) KG
--- (9) ( Compensated)
= } (!-t-f) ( I+ 0.1)')~ (t+ ~5,,P)
93
It is seen that the value of velocity error constant
is 33.3, which is greater.than 300 Hence, steady
state error at 5 rpm input will be within 1 dego The
damping ratio is Oo54o The rise time 0£ input-step
transient is 1.3 sec. The overshoot is 10%0 The settling
time is 2.1 sec. The load-disturbed transients decay
exponentially with a time-constant of 2.5 sec.
VI. NON LINEAR-COMPENSATION
The maximum degree of optimizatio:i1 ' that can be
obtained by linear comp·ensation techniques is limited·
94
by the inevitable compromises in design brought about by
inherent contradictions in the requirements for fulfilling
all the performance standards expectedo Such compromises
can be avoided only by designing an open-loop function
with a non-linear amplitude-responseo The amplitude
response may.be either continuously non..;.linear or piece
wise linearo With a function 0£ the second type, the ·
performance characteristics of the system undergoes
abrupt changes at one or more values of the error
amp1itudeo It is possible to .design for.the abrupt
non-linearities to occur at any necessary value of
error or at any given va1ue of any arbitrary function
of erroro
Even a simple_ and singl~ non-linearity of an
abrupt nature may often offer a substantial improvement
·in the optimization of a systemo The inclusion ·~f such
a non-linearity in the equalizing system of a relatively
simple control system provi~es a considerable improvement
·. in optimization at very little extra costo
·When the error . in a system is large~ a low degree
of damping offers the advantage of .a rapid recoveryo
But, as the error is reduced, an abrupt change-over to
a highly damped condition becomes desirable in order
95
to clamp down the overshoot and obtain quick settlemento
Thus, the objective in the non-linear compensation of a
position-control system is to obtain a low degree of
damping under large errors and a high degree of damping
under the low-error and the steady-state conditionso
Also~ the system gain under the steady-state conditions
should be large enough to maintain the necessary bounds
on steady-state errorso
Ao Introduction of Non-linearity as Change in Gain:
One of the simplest methods of changing the
operating conditions of a system is to change the system
gain, a larger gain n~rmally imply~ng lesser dampingo
The change in gain shifts the position of the low-frequency
asymptote of the frequency response of the system and
thereby causes a change in th~ gain-cross-over frequencyo
A change in the gain-cross-over without any change in
phase-cross-over implies a substantial change in the
degree of stability and damping of the systemo
Figure 2l(a) shows the input-step transient
response of the simple third-order type-1 position
control system optimized with linear compensationo ..
Figure 2l(b) shows the step-input transient response
96
5 ,.,.,../Su..; o., V / ol.,. "'4.·~.
· +-
--···- ··-
----4- >-- -- --\---·--
·:· · ·:....:. ·--=t::=:,__ __ ...._ _ __ .._
-+ :-+
i.A. , + )ER MARK II CHART NO. RA
(tA..) Low- GiA,N Svs r.E1'1.
Fig. 21: Input-Step-Transients or Low- and High-Gain Systems.
of the same system with the value of system gain doubled.
The latter shows a significant reduction in rise-time
over the former at the cost of lower damping. Fig. 22
shows the diagram of circuitry employed on the analogue
set-up so as to obtain a variable-gain system that tends
to follow the pattern of response of the under-damped
high-gain system for large values of error but closes
in with the pattern of response of the highly damped
low-gain system. The resulting transient response has
characteristics forming a compromise between those of
~he high- and low-gain systems. Figure 23 offers a
comparative pres.entation of the transient-response
patterns of the high-, low- and variable-gain designs.
Table III summarizes the important performance criteria
of the three cases. The variable-gain system has lesser
rise-time than the low-gain system and lesser values
of overshoot and settling time than the high-gain system.
An important observation made during the course
of the experiment is that the change in gain is most
effective in accomplishing a change in damping when the
change is introduced at the stage just preceding the
servo-motor. The change in· gain in the experimental
set-up (Fig. 22) is accomplished by changing the value
97
0.1 /.0
S,:z20V /.o
+ J=-1.. l·O
l.o T
011.,.,
/.0 faitov . -;-
Figo 22: Analogue Setup of Variable-Gain System.
----t
. =1= I ·-+---
==f=__:
- -_-\-
TI
. --CLEVELAND, OHIO PRINTED IN U.S.A.
5' ~/~ . .; a./ V/d,.. ,I.*· -· - -+----,
CHART NO. RA. 2921 32 HH.'I
{.c) ,h'~A - G'~~ Sf:ts te.m.
- --, i; -;.;._ ,..;;". j o: I v I°'. L,. .
;_ _ _
F°:l.g. 23: Input-Step-Transients of' Low-, Highand Variable-Gain Systems. ! ,
\ \
99
. -... _____ _ . _)!
.;,_,·
Noo
lo
2.;,
Jo
TABLE III
Transient Performances of the Low-, High- and the Variable-Gain Systems
Performance Low-Gain High-Gain Variable-Criteria System System Gain
~VQ+:om -
Rise Time l_o25 Sec Oo7 Sec · Oo95 Sec.
Overshoot 16 % 36 % 32 %
Settling 2o25 Sec 3o5 Sec 3o2 Sec ·Time
100
101
of the forward resistance of the 6th stage 0£ the opera
tional analogue. A pair of silicon diodes, biassed
with d.c. potential obtained from dry _cells and controlled
by the variation in potential of the input node of the
6th stage 0£ the operational analogue, are employed for
the purpose 0£ accomplishing the change in forward
resistance of the stage. A sudden input disturbance
finds the system in the high-gain condition (both diodes
conducting) so that a quick recovery is initiated. The
high rate of recovery causes a substantial change in the
potential o:£ the. input node of the 6th stage whereupon
either one of the diodes do not conduct initiating a
higher damping and quicker settlement. The system
does alternate between high- and low-gain conditions
. during the period. of transient response. A suitable
choice of the diode-bias has to be experiment·ally arrived
at so as to obtain the best possible form of response
of variable-gain system.
The experimental set-up is only a manipulation
on the operational analogue. The results merely show
that a variable-gain arrangement may be success.fully
employed for the additional improvement o.f the perfor
mance o.f a system optimized with linear compensation.
102
The actual set-up necessary for ef:fecting the variable
gain as a function of error in an actual· system is to be
determined by the nature of the parti~ular systemo
It may be pointed out, in this connection~ 1 that
the variable gain method may be employed with advantage
in . systems containing a saturating element, such as a
saturating magnetic circuito The saturating element
may be introduced at .such a point in the system as to
produce a pattern· _o:r variation o.f system gain that is
conducive to the better optimization of the system.
Bo Introduction of Non-linearity as a Change in Lag Network:
The effect o:r changing the design of a lag network
on the stability and damping of the system has been shown
in Chapter V. Bringing the corner frequencies of the
lag network closer to the poles and zeros of the rest of
the system has the effect of reducing system damping
and the rise time o:r transient responses. (Table IIo
Noo 2)o In the lag network of the form shown in Figo 13,
decreasing the value of c1 by a certain factor results
in an increase in both the corner frequencies by the
same factor~ In view of the above, an arrangement
whereby the lag network has a small capacitance for
large errors and a large capacitance for small errors
may be expected to provide a desirable form of non
linear compensation. Such a change in capacitance
at a particular value of error may be brought about
by a suitable arrangement of biassed diode circuits.
Figure 24 shows the diagram of an arrangement
103
made on the analogue set-up to satisfy the above require
ments. Silicon diodes biassed with dry cells are employed.
A low value of capacitance is in circuit when the error
is large and a rapid recovery is initiated by the under
damped situation. As the error is reduced, the larger
capacitance is connected in parallel and the system
overshoot and settling time are moderated.
In Fig. 25 are shown the forms of transient
response of (a) the overdamped system(~) the under
damped system and (c) the compromise that is under
damped for large errors and overdamped for smaller
errors. Table IV summarizes the characteristics of
these three forms of transient. It is seen that the
non-linear compromise offers a better rise time than
the overdamped system, an overshoot less than that of
the underdamped system and a settling time that is much
less compared. to that of the underdamped system and equal to
that of the overdamped system. An abrupt transition
from overdamped to underdamped condition is not clearly
104
IO o.,
IO .0 l·O
INPUT
/.o
()./
0.'1 OUTPUT
4,5V
Fig. 24: Analogue Setup of System with Variable Lag Network.
ORATION CLEV ELAND, OHIO PRINTED IN U.S.A.
I
(.cJ
- -------·--.
Fig. 25: Input-Step-Transients of Highly-Damped, Lowl:y-Damped and Variably-Damped Systems, Obtained by Changing Lag Network.
105
106
TABLE IV
Transient Performance of the Variable-Lag System
Noo Performance Highly Lowly Variably Criteria Damped Damped Damped
System System System
l. Rise Time 5.4 Sec 1. 75 Sec 2.0 Sec
2. Overshoot NIL 36 % 29 %
3. Settling 5. 2 Sec 10.2 Sec 5.2 Sec Time
107
observed in the transient of the non-linear system.
This is because the system undergoes a smooth and con
tinuous readjustment as the capacitors are charged by
the signal potential and discharged through the parallel
resistances.
The above representative designs have been so
chosen as to bring out the point that a large extent
of improvement in optimization is possible with a
variable-lag network. In a practical case, a certain
degree of optimization may be achieved by linear cir
cuitry and an additional improvement obtained by a non
linear design that offers lesser damping under large
errors.
c. Introduction of Non-linearity as a Change in Lead Network:
The possibilities of obtaining reduced rise time
of' transient by change in lead network design are investi
gated by analogue computer studies. The effects of
changes in the pole and zero of lead network with and
without change in system gain are ascertained. The
results of these studies are summarized in Table V.
An examination of Table V leads to the conclusion
that simple changes in lead network without change in gain
do not contribute to any appreciable improvement in rise
108
TABLE V
Transient Performance of the System Under Changes in Lead Network
·Rise Time Zero of Pole or
No. Lead Network Lead Network Normal Gain .Normal Gain Normal Gain x l x ·2 x 4
1 c) 1 -5 0.9 - Ose. Osc.
2. 1 10 1.25 1.1 Osc.
J. 1 20 1.1 1.2; Osc.
4o 2 5 1.2 Ose. - Osc.
5. 2 10 1.2 -1.0 -Osc. ' '.
6 • . 2 20 1.15 1.0 Osc.
timea Where accompanied by substantial increase in
system gain, an improvement in rise time is accompanied
by a considerable deterioration in system stabilityo
Also, such changes in rise time do not show much improve
ment over what was obtained by changing gain aloneo
In view of the above resu~ts, it is concluded
that no appreciable improvement in optimization c~n be
achieved by introducing a simple non-linearity that
changes the lead networko
Da Comparative Merits of the Different Methods:
The method of changing the lag network is found
to be the most suitable form of effecting non-linear
equalizationa Changing t.he lag network changes the
gain-cross-over without causing excessive change in
phase-margina Hence, an improvement in rise time is
obtained without too much deterioration in stabilityo
The common form of the lag network is particularly
suitable for this purpose since a change in the capaci
tance changes both the corner frequencies simultaneouslyo
The lead network does not lend itself so easily
to a variably camped designo An increase in gain-cross
qver frequency without too much alteration in the phase
·margin is the ideal requirement for the purpose a The
109
change in lag network achieves this end better than· the
change in lead network.
The method of changing system ~ain ~s reasonably
effective in changi~g the rise.time, without excessive
deterioration . in stabilitya This method may be of us·eful
application to systems having saturating elements a
The method o·f change in lag network has a superi-. .
ority over the method of changing the system gain in
connection with the settling time~. The variable-gain
system is found to have a longer settling time than the
low-gain systema But, the variable lag system has a
~ettling time equal in value to the highly damped systema
This behaviour may be tra~ed to the fact that the
lowly damped_ system has a larger value for the upper
~orner frequency of the lag network as compared to the
highly damped systema The effects ~n .the system due -~o -
the underdamped recovery · from. large errors decay at a
fast rate, governed by the low-time constant corres
ponding to -the lower value of lag network capacitancea
But the settlement of the overdalllped system has a sub
stantially larger time constanta Hence, the ultimate
settlement of the variably dainped system corresponds
to the highly damped systema Thus, the settling time of
the variable-lag_system is limited to that of t~e highly
da.ipped' systema
110
111
VIIo CONCLUSIONS
The more important of the conclusions arrived .
at as a result of the foregoing study on the equalization
_and optimization of a third-order type-1 position control
system may be summarized as below:
Ao Design of Phase-advanceQent:
The phase-cross-over of such a system occurs at
a frequency equal to the geometric mean of the two
non-zero poles of·the systemo For a convenient design
for good phase-margin with a single lead network, the
gain-cross-over has to be maintained in the neighborhood
of phase-cross-overo For maximum advantage of phase~
margin from a given lead.network, its corner frequencies
should be -equidistant from the gain-cross-over freque~cy
on the logarithmic scaleo A chart has been· given showing
the loci of constant .phase-margin plotted against g~in
cross-over and lead network ratioo This chart is generally
appltcable to any third order type-1 systemo Simple
expressions have also been developed for the estimation·
of the system gain required for any particular choice of
gain-cross-over and lead network ratio and for the
calculation of the corresponding value 0£ velocity-
error constanto
112·
Bo Design of Gain-advancement:
It is seen that there exists a restricted range of
values for the upper corner frequency of the lag network
within which the overall system atta.ins optimization
of transient responseo An empirical expres~ion has ·been
developed for estimating the best value of the lag network
zero in terms o.f t .he poles and zeros of the rest ·of the
system in the nei.ghborho.od of .gain-cross-overo
It has also been seen that the value of the zero
o.f the lag network determines the nature of transient
settlement under input and load disturbanceso An
.empirical relationship has been developed for predicting
a lower limit on the zero of the lag network on the
basis of the settling time requirement of the transient
perm'ormanceo
Co Non-linear Compensation:
. It has been observed that an.improvement ·over the
optimization obtained by ~inear equalization may be
effected by i~troducing a simple abrupt non-linearity . .
in the ·systemo While the method of change in gain was
found quite ·erfective, the method of changing the lag
network was found to be more e£.feotive and · desirable.o
The met.hod ·o:r changing the lead network is found to be
relatively ineffectiveo
113
D. A Design-Format:
Considering the above conclusions, the following
procedure for the equalization and optimization of a
type-1 third order system is obtained:
1. The order of magnitude of rise time of transient
required is knowu. Hence, the order of magnitude of the
cross-over frequencies of the system are determined.
(eqn. IV. 20). The basic system is so chosen as to
yield phase-cross-over at approximately this value
of frequency.
2. The amounts of phase-advancement and gain
advancement necessary are roughly estimated on the basis
of damping and steady-state error requirements.
3o The lower limit on the lag network zero is
obtained on the basis of the settling time requirementso
(eqn. V. 4.)0 Hence, the allowance for the effect of ·
lag network necessary in the design of phase-advance
ment is estimated.
4. A choice is made of the gain-cross-over
frequency and the lead network ratio from the phase
margin chart (Figo 12). The lead network pole and zero
are obtained. (eqn. IV. 4)o
5. An estimate of the best position of the lag
network zero is made on the basis of the poles and
zeros of the rest of the function in the neighborhood
of gain-cross-over. (Eqn. V.7)
6. The position of the pole of the lag network
114
is fixed on the basis of the low frequency gain necessary
for meeting the bounds on steady-state error.
7. The value of over-all system gain necessary
is estimated. (eqn. IV. 17) The limit on the velocity
error constant is verified. (eqn. IV. 1e)
B. An attempt is made to further improve the
optimization by introducing a simple non-linearity in
the form of change in gain or, preferably, in the form
of a variable lag network. The highly damp.ed part or the variably damped system is to coincide with the
linearly optimized system.
E. Further Possibilities:
An interesting possibility of the extension of
this analysis exists. A similar study on the equali
zation and optimization of a third order type-1 system
could be made by taking rate-feedback as the method of
phase-advancement and integral-reedback as the method
or gain-advancement. It may be expected that several
important and userul results may be obtained from such
an investigation. Analogue computer techniques could
be particularly userul in such an investigation.
115
116
APPENDIX I
THE TRANSFER EQUATIONS OF A SERVQ~M:>TOR
A schematic . diagram 0£ the servo-motor and load,
inclu~ing the possibility of reduction gearing, is given
· in Fig. A-lo
.The parameters of the system are define·tr as follows:
JL _ - Polar moment of inertia of load in ~lug-ft~
Jm - Polar moment of inertia of motor in slug-ft~
fL - Viscous friction of the load in ft.-lb. per
rad/sec.
fm - Viscous friction of the motor in ft.-lbo per
rad/seco
K~ - Motor torque constant in ftvlb./ ampere of
armature current.
Ke· - Motor back-emf' constant in volts per rad/sec.
~a - Resistance in ohms of motor armature.
N Ratio of reduction gearing from motor to
output shaft, Qm/Q0 ~
· Ia· - The value in amperes of the motor armature
current.
E - Applied voltage across motor armature. a
117
s,.,, Bm N=-· go
eo r t
~ig.· A-l: Schematic Diagram of Servo-Motor arid LQad ·
A. Transfer Equation Under Inertia Load:
The differential equation representing the dynamics
of the mechanical system under a constant motor field
current is:
118
When expressed entirely in terms of Q0 under the substitution
gm= NQ0 , the above equation simplifies to:
---- (2)
This represents the dynamics of a simple mechanical system
with inertia and damping subjected to a forcing function:
---- (3)
where, JeL = N2Jm plus JL and feL = N2fm plus fL represent
the total equivalent inertia and damping respectively of the
motor and load with reference to the dynamics of the output
shaft.
The differenti~l equation of the magnetically
coupled electro-mechanical system is:
{ -
In terms of the output shaft position, 9 0 , this yields
the relationship:
T :: -():..
119
(4)
(5)
Combining the dynamics of the mechanical and the electro
mechanical systems as represented by equations 3 and 5,
we obtain:
(6)
After Laplac-e Transformation and simplification, the opera-
tional transfer .:function of the over-all system becomes:
{;/, N.!t ==
Ra. ---- (7) [ ;,.t JeL + I { /e1 + N 2 Ke Kt_) ~
fl.a.
120
This is easily identified with a type-1 system of the
second order of' the form:
~ i'M. =
E tJ.. ) ( /-t- ?;.. })
---·:... where,
;11 ii 1'W?.. -
;{'°" Jet -t Ni le kc
~ Ra- Je1
~ =
f;.. jeL + N .t ke Ki
(8)
---- (9)
are respectively the g~in-constant and the iime-constant
of the given =·electromechanical system, determined, as
above, by the elect!'omechanical parameters of the syst_emo
The transfer equation of the servo-motor without
load disturbances, as stated in equa~ion I-2, is thus
establishedo
Bo Transfer Equation with an External Load Torque:
Let the externally applied load torque be L ftolbo
121
The equation representing the dynamics o:f the mechanical
system, equation 3, now becomes:
+ _[ . c1Bo JeL oU.
Combining this with equation 5, we have:
On Laplace trans:formation we get:
E . Nkt + L w Rf}..
---- {10)
---- (11)
Solution for the output position leads to the trans:fer
equation:
f)_ = 0 ---- (13)
Defining the 'normalized load torque' as the external
torque in lb.:ft. per unit o:f the total e:f:fective polar
moment o:f inertia of motor and load with respect to output
shaft,
122
we arrive at the following convenient forms for the transfer
equation of the load-disturbed system:
and
)<.~ £~ -r ~ L~
) { I+ T,.,., -f)
(14)
(15)
The transfer equations of the servo-motor with load dis
t~rbances, as st.ated in equations I-3, are thus establishedo
123
APPENDIX II
SELECTION OF A REPRESENTATIVE BASIC SYSTEJJI
The following electrical and mechanical parameters
are taken as typical or a servo-motor.
JL : Polar moment of inertia
of load
Jm : .. Polar moment o:f inertia
of motor
r1 = Viscous friction of the
load
fm : Viscous friction of the
motor
: Motor torque constant
= Motor back-emf constant
Ra : Resistance of motor
armature
N = Ratio of reduction gear
ing from motor to ~utput
shaft
2 •••• 0.90 slug-ft.
2 •••• 0.001 slug-ft.
•••• 1.0 lb.ft.sec.
•••• 0.04 lb.ft.sec.
•••• o.6 ib.ft./ampere.
0.5 volt.sec/radian.
eoo• 6.0 ohms.
oooe 10.
124
It has been shown in Appendix I that the transfer
equation of the servomotor is given by:
'9o = i'h'I EA j, { I+ ?;,, ji)
where,
Substituting the above values of the machine parameters,
/0. o. t ~ _:_.:__--~--------
6 feL + 100. o.r. ~-t
RA~ ~ k[ ?;, ::
a~ - 6/eJ +- /00· t). r;: (J. (
ZfiL + Nlie ~t
13~ ;;, = /t.. -r N~f ~ = Io + JOO. a 09- = 5 l6fl "~e.. 11. (Jr)/ ~ I o .s~ -ft! /
(). 9 -/- /OTJ.
Je, ;; + N~ J,.,., ::= -() { /0. 0.6 b ::. (/./ - .3o + 30 Oo - o.6 6. s- + (00· 0-~
z 6- I c :::: 0.1 a.nd
::: -= 3o +iJo b. ~ + /00. o.~-:. o~6
Thus, the gain-constant of the representative basic syst~m
is obtained as O.l and the time-constant of the system is
obtained as 0.1, as stated in chapter III.
125
APPENDIX IIIo
Program~ as run on the 'Royal Mcbee LGP-30'
Digital Computer System at the Computer Centre of the
Missouri School of Mines and Metallurgy using the 24o2
Floating Point Interpretive System, for the calculation .
of the open-loop frequency-response and phase-angle as
well as the closed-loop frequency-response of a servo-
mechanical. system il.f terms of the poles· and zeros of
its open-loop function:
Input Code: ;0004000,;0004000,
Location Instructions
4000 xr6300' xuo400, i0300' xm0Q00' xm0000'
4005 b0326' xpOOQOt ·xdOOQOt m0326' h0430'
4010 b0334.9. \ · h0432' h0434' xlc0002' xli0002'
4015 . le0302' xlbOOOQt xlm0000' a0430' xrOOOQt
4020 m0432' h0432' lz0016t x2i0002' x2c0005'
4025 2e0310' x2b0000' x2m0000'. a0430' xrOOOQt
4030 mo434' h0434' 2z0026t b0432 9 m0300 9
4035 'd0434' h0442' xp0000t xd0000' xn0010'
4040 m0336' xp0000t xdOOOQt b0JJ2V _h0436t
4045 h0438' xlc0002' le0302~ b0326 9 xld0000'
4050 xa0000' a0436 9 h04J6t lz0048~ x2c0005'
4055 2e0Jl0' b032q' xld0000' xaoooo, a0438' ·'
4060 h0438' 2z0056' b0436' . s0438' d0338'
126
4101 m0340' xpOOOOt xd0000' s0332t t0107'
4106 s0342' a0344' t0118' h0444t xs0000'
4111 m0442' h0450' b0444' xcOOQOt m0442'
4116 xyOOQQt h0452' a0344' . t0130' h0444'
4121 xsOOOQt m0442' ~.r0000, h0452' b0444'
4126 xc0000' m0442' xyOOQQt h0450t a.0344'
4131 t0141' h0444' xs0000' xy0000' m0442'
4136 h0450' b0444' xc0000' m0442' h0452'
4141 a0344' h0444' xcOOQQt m0442' h0450'
4146 b0444' xsOOQOt m0442' h0452' b0450'
4151 a0334' h0454' m0454' h0456t b0452t
4156 m0452t a0456t xr0000' r0442' xp0000'
4161 xmOOQQt b0326t h0440' m0330' hOJ26t
4202 b0440' . s0328' t0005• xz0000'
Data Input Code: .0004000'
Data:
Location Data Designation
4300 333 Kt
4302 l' f 00 ft 4304 4t f 01 _, Zeros
. 4306 Qt t 00 ft 0£ open-loop
4308 Qt f 00 ft function.
4Jld lt t 08 _,
4312 1' f 00 ,. '
4314 l' t 01 ft
4316 l' t 01 ft
Data: (conto)
Location Data
4318 4, /. 02 _,
4320 Qt f 00 1, 4322 Qt t 00 ft 4324 QT f 00 ft 4326 l' ./, 02 _,
4328 l' t _02 f 4330 2' /- 00 t' 4332. Qt .j. 00 /.t 4334 1' ./ 00 t' 4336 2' f 01 t' 4338 314159' /. 05 _t
4340 18' .;. 01 ft 4342 36' f 01 f' 4344. 9' t 01 ft
.f'
Designation
wINITIAL
wFINAL
127
Factor o.f Progression in w
Note: When the number 0£ zeros is less than 4, the
spaces should be .filled in as showno The No. in instruc
tions No. 4013 and 4046 should be-equal to the number o.f
zeros in the open loop .functiono
When the number o.f poles is less than B, the
spaces should be filled in as sho,-mo The Noo in instruc-
tions Noo 4024 and 4054 should be equal to the number o~
poles in the open loop .functiono
128
The values of . zeros or poles at the origin should
be represented by a very small positive value as shown
in data No 4310.
If·closed-loop-frequency response is not needed,
instructi~ns from 4103 to 4160 may be deleted and the
program becomes suitable for open-loop frequency response
and phase-angle onlyo The closed-loop response calcula
tion holds ·only for cases in which the feedback :function
is unity.
The format of .print out of results is as below:
/x tir/ ,~.foJ, f k.Gi/ ?h'1l::s-e- /l<G{ .(A.) ·
A1'7te JI+ l<G,/ · 10
A flow-chart showing the scheme of computation
is shown in Fig. A-2.
-- -/IU,/lor~ =a.
f,t<./ S,:,,s : •
+'le/
a.
b ©
129
/kt;/
Fig. A-2: Flow chart of Computations in Appendix III.
130
APPENDIX IV.
Program, as run on the 'Royal Mcbee LGP-JO'
Digital Computer System at the Computer Centre of the
Missouri School of Mines and Metallurgy using the 24.2
Floating _Point· Interpretive System, for the calculation .
of data required to plot the root loci of a servo
mechanical system, the values of the poles and zeros
of its open-loop function being known: (Maximum No.
of Zeros: 4; Maximum No. of Poles - 8)
Location. Instructions.
Input Code: ;0004000'/0004000'
4000
4005
4010
4015
4020
4025
4030
4035
4040
4045
4050
4055
4060
xr6300'
b0352'
h05QQ't
x3c0002'
b0358'
t0032'
3z0023'
·xyoooo,
xl+b0000t
a0504'
r0462t
h0504'
t0062t
xu0400'
h0462'
· x4i0002'
3e0328'
h0502'
r0462'
u0040'
a0.360t
s0500t
h0504'
xa0000'
4z0040'
uOOll'
i0.328'
p0462t
x4c0005'
b0500t
h0504'
xaOOOQt
xyOOOQt
a0502'
t0049'
4z0040'
xy0900,
b0502'
p0500'
xmOOOQt
xdOOOQt
4e03J6t
a.0356'
xJb0000'
a0502'
r0462t
h0502t
r0462t
u0057'
aOJ60t
s0504'
xd0000'
xln0.000'
b0354'
x3i0002'
h0500t
s0500'
h0502'
xa0000'
Jz0023'
xa0000'
xy0000'
a0504'
aOJ60t
xJc0002'
131
4101 3e0328' x4c0005' 4e0336' b0362' h0510'
4106 h0512' b0462' m0462' h0514' x3b0000'
4111 s0500' h0516t m0516' ao514·, xr0000'
4116 m0510' h0510' 3z0110' x4b0000' s0500'
4121 h0518' m0518' a0514' xr0000'· m0512'
4126 . h0512' 4z0119' b0512' d0510' xp0000'
4131 xd0000' b0500' d0462t xa0000' · xs0~00'
4136 xpOOOQt xmOOOQt b0462t a0400' h0462t
4141 b0500' s0400' s0356' t0147' h0354'
4146 u0149' b0404' h0J54' b0402' s0500'
4151 t0007' xzOOQQt
Data Input Code: .0004000,
Seguence of Data:
Location Data Designation
4328 l' ,', 00 ,',' zl
4330 Qt t 00 ft z2 4332 Qt f 00 f' z
3 4334 Qt ,', 00 f' Z4 4336 Qt /. 00 f' P1 4338 l' f 00 .;. P2
4340 l' f 01 ft P3 4342 l' f 01/./-' P4 4344 Qt /. 00 ft P5 4346 Qt f 00 f' p6
4348 Qt I- 00 f' P7
Seguence o.f Data:- (conto)
Location Data Disignation
4350 Qt t 00 .;., Pg
4352 l' /. 02 _, wr (Initial)
... . .·•
4354 99' - 03 _, ·Wa_ (Initial)
4356 1' f 01 ,_,
Increment on wd
435$ Qt f 00 ;.;
·4360 314159' f 05!.
4362 . lt f 00 t' 4400 5' f 01
_,
4402 2' l· 03 _,
4404 ;99, - 03 _,
4406 :rt
Note: ·Z, to Z represent the zer9s of the open loop . 4
function·of" the systemo When the No. of zeros is less
than 4, the spaces are filled in :from the beginning and
the unused spaces filled in as shown. -The numbers in
instructions No.o 4015 and 4100 should be equal to the
Noo of" zeroso p1 to Pg represent the poles.of the open ' .
loop function of" the system filled in .from the location
132
4336. Multiple poles at origin or elsewhere should be
filled in separatelyo When Noo of" zeros is less than 8, . '. .
remaining positions should be .filled in as shown. The
numbers in instructions Noo 4012 and 4102 snould be
equal to the Noo 0£ zerOSo
The format of print out of results is as below:
I When {-wd l j wr} represents a point on the root locus
on the complex plane, and K is the corresponding value
o.f system @ain and is the corresponding value of the
damping , ratio of least damped roots.
A flow-chart showing the scheme of computation
i ·s shown in Fig. A-J. ·
133
7i lfAJ~ 1. "'C",1 /di[/
.,. [ ~1 .,. (;c - t.1,tJJ YI.
Ko
fAw.' ~-(~It)
'14
+ 7r
t-----4----t,MC. ".(, ......_ ___ .,M_o""'
©
+
'J,t{,#1)
134
Fig. A-31 Flow chart or Coaputat.iona in Appendix IV.
135
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136
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137
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142
VITA
The author, VISWANATHA SESHADRI, was born on the
15th of June, 1934, at Namakkal, Salem District, Madras -
Indiao He graduated from the Little-Flower High School,
Salem, Madras - Indiao He took the Int$rmediate Course
at the Salem College, Madras - Indiao He then studied
at the PoSo~o College of Technology, Coimbatore, Madras -
India during the years 1951 - 1955 and obtained the . .
degree of' 'Bachelor of' Engineering, Electrical Branch'
in March 1955 from the Madras University of Indiao
He served at the Department of' Electrical
Engineering, PoSoGo College of Technology, Coimbatore,
Madras - ·India as Assistant Lecturer during the year
1955-56 and.as Research ·Assistant.during the year 1956-
570 Since November 1957, he is employed as the Assistant
Pro.fessor in the Department of Electrical Engine.ering,
Birla Institute of Technology, Ranchi, Bihar - Indiao·
He arrived at the UoSoAo last fall for higher
studies sponsored by a programme of technical cooperation
between the governments of India and the ffoSoAo At the
completion o.f the programme of studfes, he intends to
return to India and resume his duties at the Birla
Institute o.f Technology, Ranchi, Bihar Indiao