Research Collection

Doctoral Thesis

Nonlinear, signal adaptive and adaptive posicast control systems

Author(s): Hamza, M. H.

Publication Date: 1963

Permanent Link: https://doi.org/10.3929/ethz-a-000093179

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Nonlinear, Signal Adaptive and

Adaptive Posicast Control Systems

THESIS

presented to

The Swiss Federal Institute of Technology, Zurich

for the Degree of Doctor of Technical Sciences

by

MOHAMED HAMED HAMZA

Bachelor of Science Mass. Inst, of Tech.

Citizen of Egypt

Accepted on the recommendation of Prof. Ed. Gerecke

and Prof. Dr. P. Profos

Juris-Verlag Zurich

1963

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V.

To my Grandmother and Grandfather

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Acknowledgment

This work has been performed at the Institute of Automatic Control and

Industrial Klectronics of The Swiss Federal Institute of Technology in Zurich under

the guidance of Prof. Ed. Gerecke. The author wishes to thank sincerely

Prof. Ed.Gerecke for his supervision and for his constant encouragement

during the performance of this work.

The author is indebted to Prof. Dr. P. Pr of os for his consent to examine

this dissertation and for recommending its acceptance.

Furthermore, thanks are due to the Staff of the above mentioned institute -

especially to the chief assistant Mr. E.Ruosch - and to members of the Analog

Computer Center under the direction of Dr. H. Badr for their cooperation which

made this work possible.

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- 7 -

Contents

Chapter I Introduction

1.0.0 Brief Summary of Topics Discussed

Chapter n The Conditions for Obtaining Dynamically Equivalent

Forcing Functions

2.0.0 Introduction

2.1.0 Derivation of the Necessary Conditions

2.2.0 The Errors introduced when g (T) is

replaced by g (T)

2.3.0 The Error introduced by using different Inputs

2.4.0 Conclusions

Chapter m

Chapter IV

Chapter V

Chapter VI

Input Modification and the Improvement of System Response

3.0.0 Introduction

3.1.0 The Variable-Steps Signal Generator

3.2.0 The Variable-Frequency On-Off Signal Generator

Dual Mode Control Systems

4.0.0 Introduction

4.1.0 Theory

4.2.0 Analog Computer Circuits

4.3.0 Results

Nonlinear and Signal Adaptive Control

5.0.0 Introduction

5.1.0 Theory

5.2.0 Analog Computer Circuits

5.3.0 Analog Computer Results

Optimum and Quasi-Optimum Control. First-Order Systems

6.0.0 Introduction

6.1.0 Theory

6.2.0 Experimental Study

11

11

12

12

13

17

25

35

38

38

39

47

53

53

53

57

59

74

74

75

77

81

94

94

95

99

Chapter VII Optimum and Quasi-Optimum Control. Second-Order Systems 102

7.0.0 Introduction 102

- 8 -

7.1.0 Compensation using Pulse Modulation 103

7.1.1 Introduction 103

7.1.2 Statement of the Problem 105

7.1.3 Solution 105

7.2.0 The Impulse Modifier 117

7.3.0 The Pulse Modifier 124

7.4.0 Posicast Control 130

Chapter Vm Posicast Control Applied to Higher-Order Systems 142

8.0.0 Introduction 142

8.1.0 Third-Order Systems 142

8.2.0 Fourth-Order Systems 152

8.3.0 Fifth-Order Systems 152

8.4.0 Conclusions 155

Chapter DC Adaptive Posicast Control 157

9.000 Introduction 157

9.1.0 Signal-Flow Diagram 158

9.2.0 Theoretical Analysis 163

9.2.1 The Transient and Steady State Performance of 165

the Adaptive Control System

9.2.2 The Stability of the Adaptive Control System 171

9.3.0 Analog Computer Circuits 173

9.4.0 Results 176

Chapter X Concluding Remarks 182

10.0.0 Some Remarks and Suggestions for Future Work 182

Appendix A 184

Appendix B 199

Zusammenfassung 201

Bibliography 203

- 9 -

Notation

r(t) input

c(t) output

e(t) error

D differential operator unless otherwise stated

s Laplace operator

»0(t) a unit impulse

6(t) a Dirac delta function

M(t) a unit step

«.2w a unit ramp

u^t) a doublet, that is, the derivative of a unit impulse

R(s) Laplace transform of r(t)

C(s) Laplace transform of c (t)

g(t) system's impulse response

G(s) Laplace transform of g (t), that is, system

transfer function

Fig. figure

eq. equation

sec. seconds

Hz Hertz, that is, the I.E.C. equivalent of cycles per second

[12] reference "12" given in the bibliography

*»« modified input

'«.» modified input when a sampler is used to form the modification,

For simplicity in most cases merely r (t) is used

cr(t) required output

- 10 -

SDCS sampled-data control system (s)

T Sampling period

h pulse duration

t time. The units of time are seconds unless otherwise stated

C computer

CO controller

R relay

u (t) the output of sampler having a finite pulsewidth when its input

isu(t)

u*(t) the output of a sampler having an infinitesimal pulse-width when

its input is u (t)

- 11 -

Chapter I

Introduction

1.0.0 Brief Summary of Topics Discussed

Under certain conditions different inputs to a linear system produce approxi¬

mately the same outputs. In chapter n this problem is studied and suggestions are

made as to how signals should be chosen to be dynamically equivalent, that is to

produce approximately the same responses.

In chapter HI a variable-frequency on-off signal generator and a variable-

steps signal generator are described. These signal generators are used in chap¬

ters VI, VII and Vffl.

By allowing a parameter of a control system - such as damping or sampling

duration - to have two values and by having a computer control the instants of swit¬

ching from one value to the other, improved system step response may be achieved.

This is illustrated in chapter IV.

In chapter V nonlinear amplifiers and signal adaptive samplers placed in

either the system forward or feedback path are shown to improve the step respon¬

se of control systems.

In chapters VI, VH and Vm the posicast method of control is discussed and

is extended. Studies of first to fifth-order systems are given.

Li chapter K a method is introduced for achieving adaptive posicast control

for second-order systems. Analog computer results illustrating the effectiveness

of this method are presented.

Finally, in chapter X some conclusions and suggestions for future work are

given.

- 12 -

Chapter II

The Conditions for Obtaining Dynamically Equivalent Forcing

Functions

2.0.0 Introduction

In control problems it is of major importance to be able to express the sig¬

nals in the system in a convenient mathematical form. If the input signal to a linear

system is of a form amenable to analysis, such as a sinusoid or a step function,

then the output to that system can be usually easily determined. If the input signal

to the linear system is not of the familiar type, but is for example like the signal

shown in Fig. (2.0.1) curve "a", then to determine the system response, the input

signal can be approximated during appropriate time intervals by familiar functions

such as step functions, ramps or exponentials, see Fig. (2.0.1) curve "b", and by

superposing the responses to the familiar functions, the appropriate delays being

taken into account, an approximate system response may be obtained.

The disadvantage of this method is that it is sometimes very tedious and

time consuming due to the possibility of having to deal with many different time

functions. An alternate approach which is simple and which proved to be very use¬

ful in solving other types of problems, consists of determining another function

which is amenable to analysis and which is dynamically equivalent to the first. Thus

the only similarity between the first signal and the second one is that they be dyna¬

mically equivalent. The conditions under which two inputs to a dynamic system

produce essentially the same output are expressed by the Principle of Equivalent

Areas which may be stated as follows:

Fig. (2.0.1) An unfamiliar-looking Signal and a possible Approximation of it

- 13 -

The input function r,(t) and r,(t) to a dynamic system are dynamically equi¬

valent if

JB r,(t) dt = |n r,(t) dt (2.0.1)

Tn-1 Tn-1

where (T -T -) is a time interval smaller than half the smallest of the input and

dynamic system significant time constants, "n" is a number which takes on all

values sufficient to cover the total time duration of interest.

The inputs r. (t) and r„(t), when the above conditions are satisfied produce

essentially the same outputs respectively. The approximation depends upon the ti¬

me interval (T -T ,) and can be made as good as desired by choosing (T "T ,)n n~x It— n~A

small.

2.1.0 Derivation of the Necessary Conditions

To derive the conditions for which two signals are dynamically equivalent,

let r (t) represent the input to a given linear continuous system and let c (t) repre¬

sent the corresponding output. Choose the time origin so that

r(t) =0 for t -to (2.1.1)

Suppose r(t) and c(t) are related as follows:

<££&)+ P

dn-1c(t)+ + p JMO

+ p c(t) _ r(t)

(2.1.2)

where "n" represents the order of the differential equation and is constant. P.., P„

...P are constants and together with "n" depend upon the characteristics of the

linear system.

Equation (2.1.2) may be written in the abbreviated form

Dc(t) = r(t) 0 i t < cd (2.1.3)

where "D" is the differential operator of the system and is given by

(2.1.10)dXr(X)g(t-X)j=c(t)

or

(2.1.9)dTr(T)u^t-X)g(t-X)f=c(t)

obtainedisfollowingthe5),(2.1.equationin(2.1.8)equationSubstituting).XG(t-

ofargumentnegativetheforX)G(t-ofextensioncontinuoustheisX)g(t-where

(2.1.8)gft-TJu^t-T)=G(t-T)

aswrittenbemayX)G(t-Therefore,later).discussedbewill(this0<tforzero

isG(t)systemphysicalainthatfoundisit5),(2.1.equationtoReturning

oo.<•X40intervalhalf-infinitetheindefinedion

funct¬continuousanyisandfunctiondeltatheforfunctiontestingtheis)$(Xwhere

o

(2.1-7)=4>(t)dt(X)*6(t-X)J

follows:asdistributionofsensetheinned

defi¬iswhichfunctiondeltatheisT)(t-6andsystemtheofoperationdynamicthe

describeswhichconditionsinitialzerowithoperatordifferentialtheis"D"where

(2.1.6)oo<t«06(t-X)=T)DG(t,

equationdifferential

followingtheofsolutiontheissystemcontinuousaoffunctionGreen'sThe

"D".operator

theoffunctionGreen'stheisT)G(t-convolution.realaofformtheiniswhich

(2.1.5)dTr(T)G(t-T)J

=:(t)o

;

oo

obtainedis5)(2.1.equationc(t),for(2.1.3)equationSolving

'vndtn-1jxn-11j.n

(2.1.4)+Px«+P...+=--

P,+-—

ddn_1,,

dn

-14-

- 15 -

which is the more familiar form of the convolution integral. The Green's function

g (t- X) may be considered as a weighting function and represents the impulse res¬

ponse of the system for the impulse occuring at t = T .The impulse response of

a physical system is zero for all negative time since in case of a physical system

no output can be obtained before an excitation is applied. Consequently, the state¬

ment that for a physical system G(t) is zero for t < 0 is a valid one.

Consider as a possible impulse response to a given linear system the function

g(T) shown in Fig. (2.1.1)*. Suppose that g(T) is approximated by the stepwise

function g_(T) shown in the same figure. Note that the function g_(T) is constant

gW gB(*)

a-g(*)

b=gp(r)

Fig. (2.1.1) The Impulse Response g(T) of a given. Linear System and its

Stepwise Approximation g_(T)

during any interval T,to T Assuming that such an approximation is valid

(the validity will be discussed later), then c(t) may be written as

c(t) = J gJt-T) r(T) d (2.1.11)

Assume further that the time "t" falls at the end of one of the steps of g (T), if not,

then choose the steps in such a way that "t" does fall at the end of one of the steps,

say the m step, that is

where m = 0,1,2 ...,n (2.1.12)

*) The derivation that follows is similar to that presented by Andeen, however,it is more general and it is developed further.

- 16 -

therefore

c(T) = JmgD(Tm- T)r(T)dT (2.1.13)

The integral from zero to T may be broken into a sum of integrals over the inter¬

vals T* to T

,where "n" varies between zero and "m". Therefore,

n-l n

m T

c(T_) = Z In Sjtm - T) r(T) dT (2.1.14)n=o T

n-l

°p* m

During any interval T,to T

,g ( T) is a constant and is equal to g ( T ). see

n-l n p p n

Fig. (2.1.1). Therefore c( t ) may be written as

c(TB)BIS(Tm-^/ r(t)dT <2-1-1«

Defining

n=°P "

t'n-1

F(V = |n r(T) dT (2.1.16)

Tn-1

c (T_) reduces tom

c(Tm) = Z Bp(Tm- tn)u(Tn) (2.1.17)n=o

From equation (2.1.17), it is seen that c( T ) depends upon u(T ) which is a num¬

ber and not on the exact shape of r(T). This means that if the input functions r,(T),

r2(T), r_( T), which may all be different, satisfy the relation

^Xr) = r rl(T) dt = /" r2(T) dT

Vl Vl

= ;n r3(T) dT (2.1.18)

Tn-1

where "n" takes on all values sufficient to cover the total time duration of interest

and where (T - T ,) are small time intervals - their duration to be determinedn n-l

later - then these inputs will all produce essentially the same outputs. The above

- 17 -

derivation holds provided it is valid to replace g(T) by gD(T).

2.2.0 The Errors introduced when g (t) is replaced by g0(T)

To construct g (T) from g(T), g(T) is assumed to be a signal which is

being sampled at times T T T, etc., and that the sampler is followed by

a zero order hold circuit, see Fig. (2.2.1).

u.(*)>-9fr) Tn^gfo)|

9pW

Fig. (2.2.1) Signal-Flow Diagram showing how g (T) may be obtained from g (T)

u0(T) A unit impulse

D The given linear continuos system

H A zero-order hold circuit

Tn The sampling period, n = 0,1,2 ..

g(T) Output of linear system

g*(T) Output of sampler

gp(T) Output of hold circuit

From Fig. (2.1.1) it is obvious that:

a) The smaller the time interval (T - X .), the better g (f) approxima¬

tes g(T).

b) For a given time interval, g (T) is a better approximation to g (X) the

smaller the slope of g ( X ) is in that interval.

Due to points a) and b), the aim will be to try and find the largest time inter¬

val which does not produce an appreciable error. Li a sense, the largest time inter¬

val is unique. For this reason and also in order to simplify the mathematics it will

be assumed that the time intervals are constant, that is

rn- Vl = T (2.2.1)

- 18 -

where T is a constant and has the units of time.

Under such conditions, Fig. (2. 2.1) reduces to Fig. (2.2. 2) in which the samp¬

ler closes periodically with a period T and the zero order hold circuit has as a

transfer function [17, 28].

-sT

H(s)1 - e

(2.2.2)

To determine the frequency response of this hold circuit "j«j " is substituted for "s"

in equation (2.2.2) resulting in

H(jco) =

B-JwT

]CO

tuTsm -5"" , ,T

H(jw) = T m% exp (-j^i)

(2.2.3)

(2.2.4)

u.W J

Fig. (2.2.2) Signal-Flow Diagram showing how g (t) may be obtained from g(t)

u0(T)G(s)

H(s)

T

g(t)

g*(t)

gp(t)

A unit impulse

The given linear continuous system transfer

function _

1 - eA zero-order hold circuit. H (s) =

s

The sampling period and is constant

The impulse response of the given linear system

The sampled impulse response

The stepwise approximation of g( T)

The amplitude and phase of H (jw) are plotted in Fig. (2.2.3). The frequency

response of the zero order hold circuit is low pass with full cut-off occuring at fre¬

quencies of 5r Hz, where "n" is an integer.

- 19 -

With the aid of Fig. (2.2. 2) and equation (2. 2.2), the Laplace transform of

g (T ) which is G (s), may be written as

i"sT

Gjs) = i^ G*(b) (2.2.5)P °

where G*(s) is the Laplace transform of g*(T), the sampled response of g(T ),

and is expressible as

G*(s) =i Z G(s + j^) (2.2.6)

n=-oo

see [17, page 9],

Equation (2.2.6) is of the infinite series form and indicates that the sampler

produces high frequency components.

Assume that the Fourier spectrum of g (t) is that shown in Fig. (2.2.4). It

contains negligible energy outside of some low frequency region o> . Most spectrums

in the real world have a spectrum of this type, although the location of u for a gi¬

ven system depends upon what we mean by negligible energy. With the aid of equa¬

tion (2.2.6), the spectrum of g*( T) could be obtained. This spectrum is plotted in

Fig. (2.2. 5). By inspection of g*( T) it is seen that the margin between shifted rep¬

licas of the original spectrum is given by

margin = co - 2 wo (2.2.7)

- 20 -

v 90

\>,0

-90°-

-180°

-270°

\\\\\\\

CO

Fig. (2.2.3) Gain and Phase Characteristics of a Zero-order Hold Circuit.

21

*- CO

Fig. (2.2.4) The Fourier Spectrum of g(T)

T|G(jco)|i

Fig. (2.2.5) The Fourier Spectrum of g*( T)

2 "Hto = -=- where T is the sampling period

u is the bandwidth of the given linear system

- 22 -

If such a margin exists, that is, if the individual spectral pulses do not over¬

lap, then the spectrum of g*( T ) contains no less information about the original sig¬

nal than does the original spectrum of g(T). In other words, the original signal

g (T) may be recovered from the sampled signals g*(T). This may be done by pas¬

sing the sampled signal through a frequency filter that tends to reject frequencies

above to,but passes those frequency components below u) without distortion.

The upper limit on the sampling period T so that the original signal may be

still recovered from the sampled signal is T« where

T, = |-5- and where ^,

= 2U (2.2.8)1 to

.Jsi o

v

This result is stated in Shannon's sampling theorem [16]. In Fig. (2.2.6) G*(jco)

is plotted for T = T±. In Fig. (2.2.7) a sketch of H(jw) for T = T- is given. With

the aid of equation (2.2.5) and Fig. (2.2.6) and (2. 2. 7), the spectrum of g ( T) for

T = Tj is easily obtainable. G (jw) for T = Tj is shown in Fig. (2.2.8) curve "b".

Let e (t) represent the error introduced by using g (T ) instead of g (T), the input

being r,( T). Therefore,

t t

e(t) = } g(t-T)r1(T)dT -f g (t-T) r^T) d t (2.2.9)o o

p

Taking the Laplace transform of both sides one obtains

E(s) = R^s) G(s) - Rj(s) Gp(s) (2.2.10)

or

E(s) = Rj(s) [G(s) - Gp(s)] (2.2.11)

23

Ti

-i 1- . CO

-2cJs -ws Hj 0 CJ COo s

2coc

Fig. (2.2.6) The Fourier Spectrum of g*(T) for T = Tj

|H(j<40(«)

Fig. (2.2.7) Gain and Phase Characteristics of the Zero-order Hold Circuit

for T = T,

T, = —— where to is the cutoff frequency of the linear system to is the samp-i to„ o

...o

o ling frequency

24

|G(jco)| A|Gp(jco)

Fig. (2.2.8) The Fourier Spectrum of g (T) and of g (T)

a) The Fourier Spectrum of g( X)b) The Fourier Spectrum of g ( T)

W, 2 co„

-1

'G(jw)| if |Gp(Hl

-Q25

i a

Q25

*r7^/77^^ermrt^ws

Fig. (2.2.9) The Fourier Spectrum of g(T) and of g (T) for to = 4o>^~^~"~"~",^—^~— p SO

- 25 -

If equation (2.2.11) is written for values of the complex variable "s" on the

j-axis, equation (2.2.12) is obtained:

E(jco) = Rjfjio) [G(joo) - Gp(jco)] .(2.2.12)

where E(jco), G(jco) are the spectra of e(t), rJ T), g(T), and g (T) respectively.

The shaded area in Fig. (2. 2.8) represents I G(jco) I - IG (jco) I. It is seen that

most of the spectral energy of G (jco) is concentrated in the low frequency band

which extends from -co to co . It is also noticed that G (jco) contains less energyo o p

thanG(jw) In that band. In Fig. (2.2.9) G(juj) and G (jco) are plotted for co =

p s

4to.The shaded area in Fig. (2. 2.9) which also represents IG(jco)I - IG (jco)I

is seen to be much smaller than that in Fig. (2.2.8). Since a high value of oo cor-s

responds to a small sampling period T, the result obtained above checks with the

previous statement that the smaller the sampling period the better g (T) approxi¬

mates g( IT).

2.3.0 The Error introduced by using different Inputs

Suppose the signals shown in Fig. (2.3.1) are used as inputs respectively to

a linear system having the impulse response shown in Fig. (2.3.2). The spectra

of the corresponding outputs is then given by

CjOco) = RjpOco) G(jco) (2.3.1)

These spectra are plotted in Figs. (2.3.4) and (2.3.3). On examining these spectra

the following may be concluded: '

*) Note: These conclusions are valid for the specific choice of g(T) made here.

For other choices of g (T) which are met in practice, these conclusions

will-in most cases be valid.

26

^(t)

"T*-^

>^

-*-t

2T, 3T, AT! 5T,

ra(t)

7.5

3Tj ATj

(a)

*»t

- 27 -

r„(t).

21 -i

3Ti

(d)

--»-

3.25Tj IT,

rg(t>/ l

60+|

3Tj

(g)

3.125Ti

r.(t>,

20

(h)

375Ti

'f"'*

7,5T,'1 ,\

15

3Tj

(f)

371 «Ti

r-.lt) i

(i)

3.5Ti

Fig. (2.3.1) Some Dynamically Equivalent Signals

The units used have the property that 7.5 Ti are equal to one unit of area.

4T„

/*

rt It) dt = 1

3T«'p

p = a,

- 28 -

|g( jo.) | K

\/me

/o,6-

1 0,4-

/ 0.2-^

U3

-3a»0 -2(Uo -6J0 0 <0o 2 ou0 3 0jo

Fig. (2.3.2) The Spectrum of a Linear System

a) Signals which have equivalent areas and which have the same duration give

rise to outputs that have similar spectra. This may be seen from the curves of

CJjeo) and C(jto) - where p = a,b, c, d,e - which are shown in Figs. (2.3.3)

and (2.3.4).

b) If two signals having equivalent areas are allowed to have a duration ran-

2 Ttging from zero (for an impulse), to T-, where T, = -t-t— and where u> = 2 w

,

0l> being the bandwidth of the dynamic system, then the maximum error - where

by the error here is meant the difference between the spectra of the corresponding

outputs - occur for the case where one of these signals is a unit impulse and the

other is a rectangular pulse, see Fig. (2.3.4) "a" and "f".

c) If in b) the duration of the rectangular pulse is decreased, then the error

is also decreased, see Figs. (2.3.3) and (2.3.4) spectrum "g" and "f".

According to conclusion a), if a signal is given and if it is required to obtain

another signal dynamically similar to it, then it is advisable to let the new signal

have the same duration as that of the original signal.

The analysis up to now was concerned with the determination of the allowable

duration of T. The duration of T is considered to be permissible if it results in a

small error. In the above, the error was interpreted as meaning the difference in

magnitude between the output spectra. Such an error does not give much information

about the time domain output error. A convenient method for calculating the latter

is as follows:

- 29 -

Cv(jw)

0,2S 0,375

Fig. (2.3.3) The Spectrum of Cj (T) for p = c,g,h

Cuj,<]«>

H

m

be'\k

0.6*K\!\

*'

0.2

_^Jjf01 aos an CP75 as

Fig. (2.3.4) The Spectrum of c(T) for p = a, b,e,d,f,i

- 30 -

Suppose rAt) and r,(t) are two dynamically equivalent signals - by dynamical¬

ly equivalent is. meant that they satisfy equation (2.3.2)

nT nT

f rt(t) dt = / r,(t) dt = F(nT) (2.3.2)(n-l)T

*(n-l)T

z

where T is a small time interval and "n" is an integer which takes on all values

sufficient to cover the total time duration of interest, u (nT) is a constant and may

be different for different time intervals. If rAt) is used instead of r„(t) as the in¬

put to a linear system having the impulse response g(t), then the corresponding

output Cj(t) will differ from c„(t) by an amount e (t) called the time domain error

and is given by

e(t) = Cl(t) - c2(t) (2.3.3)

Equation (2.3.3) may be written as

t t

e(t) = / g(t-T) r^T) d - / g(t-T) r2(T) dT (2.3.4)o o

or equivalently as

e(t) = | J [g(t- T) - gp(t- T)] ri(T) dT + | gp(t- T) r^T) dt|

- | I feft- T) - gp(t- T)] r2(T) dT + J gp(t-t) r2(T) dtl (2.3.5)

Making use of equations (2.1.10) to (2.1.17) and of equation (2.3.2), e(t) may be

written as

fmTe(mT) = j J [g(mT - T) - gp(mT - T)] r^T) dT

m 1 [mT

+ Z gJmT - nT) u(nT) [ - i j [g(mT - T) -

n=op J I o

gp(mT-T)] r2(T) dt

-concluded:

bemayfollowingthe(2.3.7)Fig.and(2.3.5)Fig.(2.3.7),equationFrom

(2.3.7).Fig.intabulatedareresultsThe(2.3.6).

Fig.inshownT)(g-(T)gtheusingsignalsequivalentoftypesvariousforerror

periodonethedeterminingsuggestedThis(2.3.6).Fig.inshowncategoriestheof

oneunderfallperiodoneanyduring(t*)]-g[g(T)ofshapestheFurther,fixed.

are]r2(T)-[r^T)and]T)(mT-g-T)[g(mT-period,oneanyDuring

r..(T).oneequivalentdynamicallyaofchoicethe

r2(T)andsignaloriginaltheupondependr,(T)]-[r-CC)(2.3.5).Fig.inas

givenare(T)gand(T)goncedeterminetoeasyis]1)(mT-g-T)(mT-tg

t)].r2(-T)[r^by)]X-gp(mT-T)-[g(mTmultiplying

byobtainedcurvetheunderareathedeterminetois-(2.3.7).equationtoding

accor¬-procedurethegraphically,errorthedetermineToshown.areV)(gtion

approxima¬stepwiseitsand(T)gresponseimpulsesystema5)(2.3.Fig.In

simplicity.itsforchosenwasmethodgraphicalThe(2.3.4).

equationthangraphicallyevaluatetoconvenientmoreis(2.3.7)Equationb)

(2.3.4).equationusingthan(2.3.7)

equationusingspottoquickerareerrortheofcharacteristicstheofSomea)

that:are(2.3.4)equationthanrather

(2.3.7)equationemployingerroroutputthedeterminingofadvantagesThe

^o

(2.3.7)r2(T)]dT-(mT-T)]tr^T)g-[g(mT-T)/=e(mT)mT

tosimplifieswhich

pn=o

(2.3.6)\ji(nT)nT)

-S_(mT51+

m iji(nT)nT)

-31-

- 32 -

2Cff~r

Fig. (2.3. 5) g (T ) the Impulse Response of a Linear Continuous System and its

Sepwise Approximation g ( T )

a) The sign of the time derivative of g ( T) affects the sign of e (mT).

b) The location of r1( T ) and r„( T) within the interval T affects the sign and

the magnitude of e(mT), Fig. (2.3. 7) "1" and "5".

c) The form of r,(T) affects the sign and the magnitude of e(mT), Fig. (2.3.7),

"7" and "9".

d) The shape of g(T) affects the magnitude of e(mT).

e) If the position of rAT) can be chosen arbitrarily within the period T, then

it is possible to determine a location for which the error e (mT) is zero, Fig. (2.3.7)

"20" to "26".

f) If the position of tA t) is chosen such that the "center of gravity" of r..( T)

and r,(T) coincide*, then for the case where the system impulse response is linear,

Fig. (2.3. 7), the one period error produced by using rAT) instead of r„(T ) will be

zero.

Further deductions can be made from Fig. (2.3.7), however for brevity, it is

left to the reader to search for the ones he is most interested in.

*) rj(T) and raft) are treated as If they were metal sheets. Their center of gravityis found using the same procedures employed in mechanics.

- 33 -

la)

g(r)

2.0

1,0

"9pW

(q-DT qT

(b)

g(t)-gp(r), I

2,0

toT

i—*-

U

-1.0

* — T—-^—»

2,0(r-l)T rT

(c)

t r

6.0

5.0

4.0

3.0

2.0

1,0T

0

(k-1)TT »

kT

Fig. (2.3.6) The General Forms of [ g ( T ) - gp( T ) ] during any one Period.

q, r and k are integers

- 34 -

<iM.r2hr)3D —i

x- ;

.ote'-^'

f\M.r2M

-F=j-

n

jjWrglT)

q,!*)so

VT) 80^h)

«>,* ^t \ * w*

0 T Z >1—""l "5 T

46 43S 800

-186 949 1500

-232 524 600

-3B9 1249 1800

-28 -384 -900

-389 49 -1800

-219 -275 -1S00

-242 -424 -1860

23 148 360

24 277 600

-209 B01 1134

-412 noi 1440

-412 -99 -2160

28 554 1200

- 35 -

r,<Uij|TT

,p*M«»tf*l

<24>

(25)

-M -256 -600

137 122 BOO

-4* 675 1200

B6 326 920

-6 (07 770

134 -211 0

75 -124 0

33 -26 0

0 -395 770

D -57. 660 720 3060

146 0 240

193 0 -579

103 206 340

65 63» 1350

-SI -S33 -1200

Fig. (2.3.7) A Tabulation of the one period error which results when rj(T) is used

instead of r£( T) as the input to the system having the impulse response

q(- X) shown in the columns to the right. q(- T) = g(- X)-gp(- T).The values in the columns should be multiplied by _3L_

.The results

obtained after the latter multiplication has been 600

represent the relative values of the one period error,

represent rj(T)represent rg( T)

performed

- 36 -

Returning to the original problem of determining the error introduced when

rAX) is used instead of r„( T ) as an input to the linear system which has the im¬

pulse response g(T), the procedure to be recommended, as a result of the studies

which have been done, is the following:

Choose a period T such that T < l/2 T where X.

is the smallest sig¬

nificant time constant of the system and the input signal. Construct a convenient

rj(t) such that

nT nT

J rj(T)dT= / r,(T) dT (2.3.8)(n-l)T

X(n-l)T

l

where "n" is an integer and "T" is the period mentioned above. Construct g (T),

then obtain [g(T) -g (t)l.

Determine

mT

| [g(mT-T) . g (mT-T)] r^T) dT = fj(mT) (say) (2.3.9)o

v

and similarly determine

mT

J [g(mT-T) - g (mT- t)] r2(T) d = f2(mT) (say) (2.3.10)o

v

The original system response c„( T) can be expressed as

c2(T) = Cj(T) - e(T) (2.3.11)

The output error e(mT), with the aid of equations (2.3.7), (2.3.9) and (2.3.10),

is given by

e(mT) = ft(mT) - f2(mT) (2.3.12)

Since the sampling frequency is high compared to the significant frequency

of the input and the highest slgnificantfrequencyof the spectrum of the dynamic system,

then it should be possible to determine e (T) - the required error - graphically by

joining the points determined by e (mT).

- 37 -

The incentive in introducing rAX) was that the response c-(T) is easier to

determine using standard methods than c,( T). Consequently, once cAT) is deter-

minded, then c„(T) may be obtained by substituting Cj(T) and e(T) in equation

(2.3.11). In practice, usually it is required to determine only the maximum per¬

centage error so that e (T) need not be determined, but merely a few points of

e (mT) - those which result in large percentage errors. In finding these points,

Fig. (2.3.7) can be of great help.

2.4.0 Conclusions

The input signals r-(t), ... r. (t) to a linear continuous dynamic system are

dynamically equivalent, that is they produce essentially the same outputs, if

rk(t) dt (2.4.0)T T T

r r^t) dt = /" r2(t) dt =...

= rT

<T

1T

1n-1 n-1 n-1

where (T -T ,) are time intervals which are smaller than half the smallest of then n-1

input and dynamic system significant time constants respectively, "n" is an integer

which can take on all values sufficient to cover the total time duration of interest.

The approximation depends upon the time intervals (T -T ,) and can be made as

good as desired by choosing the time intervals small. The maximum percentage

error produced by using r.,(t) as an input to the dynamic system instead of r,(t)

can be estimated quickly using the method suggested in section 2.3.0.

If the position of the r.,(t) pulses can be chosen at will within their respective

periods, then it is possible to determine a location for these pulses such that the

error produced by using r. (t) instead of r„(t) is a minimum.

The concept of dynamic equivalency of signals should not be underestimated,

but should be always kept in mind. B has already been successfully used in the stu¬

dy of nonlinear systems and will be used in this dissertation.

- 38 -

Chapter III

Input Modification and the Improvement of System Response

3.0.0 Introduction

When it is required to synthesize a linear control system, one finds that

several methods exist [16]. By the aid of these methods, one can compensate the

system under consideration in such a way that it is stable and satisfies certain ar¬

bitrary performance requirements, such as rise-time, peak overshoot, etc.

Linear control systems - as designed by conventional methods - have some

serious limitations. These are mainly, that the maximum available power is usual¬

ly used only for a small fraction of the time, leading to a low efficiency and a slow

response time. Also, the transient response of linear control systems is theoreti¬

cally infinite in duration. By deliberately inserting nonlinear components in control

systems or by operating linear components in nonlinear regions, it has been found

that improvement of system response may be achieved [34]. However, with the

development of modern technology, requirements on system performance became

more and more stringent. This led control engineers to direct their work towards

optimization [5]. The aim was no longer to design a system that works well, but

to design the system that works best. As an example, it has been found that as far

as the speed of response is concerned, the on-off servomechanism is the optimum

one.

In the following sections it is shown that several methods exist for modifying

the inputs to linear systems such that the corresponding responses have short rise-

times and zero overshoot. The input modifications vary, some are simple to instru¬

ment, while others require the use of complex circuitry. For simplicity, the open

loop case is studied. The controllers determined, however, can be - if requested -

incorporated into the closed loop, using for example, the block diagram manipula¬

tion techniques described in reference [31].

In the next two sections a description of the apparatus used is given. In the

following few chapters, first to fifth-order linear open loop control systems are

studied.

- 39 -

3.1.0 The Variable-Steps Signal Generator

To determine what modification the step input to a linear control system

should undergo in order that the corresponding response have better qualities than

the response otherwise obtainable, it is convenient to build a signal generator whose

ouput may take a large variety of forms and to test the response of the system un¬

der investigation to those signals. For convenience, the output of this signal gene¬

rator should be of a stepwise nature, because then the compensator that modifies

the step input to give such an output can be synthesized using delay elements or in

many cases by incorporating a computer controlled switching device in the path

of the step input. This type of compensation is discussed in chapters VI, VII and

vm.

A signal generator having the characteristics described above has been built

by the author. This signal generator will be given the name "variable-steps signal

generator" and for convenience, will be referred to simply as the VSSG.

The signal flow diagram of the VSSG is shown in Fig. (3.1.1). This VSSG

consists of an astable multivibrator M* which starts to oscillate on closing the

switch S and reopening it. The multivibrator M. provides the clock signal x-

which is a square wave. The frequency of this square wave may be chosen as one

wishes to lie anywhere between 0,5 cycles per second and a few kilocycles per se¬

cond by simply changing the values of two specific condensors in the circuit of the

astable multivibrator. Consequently, if large enough condensers are used, this

signal generator will be suitable for providing the input signals to systems simula¬

ted on an analog computer. By introducing x- into an inverter circuit, the signal

x, is obtained. By means of differentiation and clipping, the trains of negative

pulses s. result. s1 is then made to trigger a monostable multivibrator M„, which

is originally in the "on" position, giving the signal sg. The duration "hj" of the ne¬

gative pulses of the signal s- depends upon the value of an RC time constant and

thus may be chosen as one wishes by varying R or C or both R and C. In the circuit

built by the author, the time constants of the monostable multivibrators are varied

by varying the values of the resistances. The signal s,. is then fed into an inverter

giving s„. s10, S-- and s.., are obtained from sg, s«0, s.j, respectively in the

same way sg was obtained from x,, see Fig. (3.1.1). The signals sg ... s.,, are

then fed into the inverters L.... L, giving s«3 ... s-g. The voltage levels of the

- 40 -

-JZ^L.

LUn„ J; Ijf

-t\ 'D '6 'D

•a *n • -*p

*a a a

IB' !fiCD

J5.

ill

of

a

J*

K

m

JiP

- 41 -

Ss

Ml

Ir. h

cr • •c4

M2. ..M5

pr ..P6

N

El

xl'x2

sr.,.s4

s5..,.s8

v- •s12

s13- • • s16

er. .e6

e7

e8

T

h,.. • h.

A switch which when pressed then released, gives an impulse that

starts the astable multivibrator to oscillate

Astable multivibrator

Inverter followed by an emitter-follower

Converters for the derivation of the signals s1 s. from the sig¬

nals x2,Xg ... s-., respectively through differentiation and clipping

Monostable multivibrators

Potential dividers

A diode "or" circuit designed for operation with negative input pulses

An emitter-follower having a very low output impedance

The clock square wave x- and its inverse x.

Four trains of pulses derived from the signals x«, sg... s^- respec¬

tively through differentiation and clipping

The output pulses from the monostable multivibrators M, ...Ir¬

respectively as the result of the latter being triggered by the train

of narrow pulses s. ... s.

The inverse of the signals s-... sg respectively

The inverse of the signals s„ ... s-2 respectively

Signals proportional to Sjg • • • s«» respectively

The output of the "or" circuit

The variable-steps signal generator output

The dotted lines indicate that these leads may or may not be present,

as one wishes.

The period of the clock square wave x.

The pulse-widths of the signals generated by the monostable multi¬

vibrators M_ ... Mj. respectively

- 42 -

clock signals x. and x2 as well as that of the signals s13 ... s,g assume only two

values, one close to zero and the other a negative one. The latter voltage depends

upon the potential of the power supply used. To obtain signals that have a wider

scope of amplitude variation than s«g ... s1G, x, and x.,, use of potentiometers

is made. These potentiometers are denoted by P.. ...P- and their outputs are

e., ... e~ respectively. By introducing any combination of the signals C. ... efi

to the inputs of the "or" circuit designated N, the output signal e„ is obtained.

Clearly an extremely large number of variations of e„ is possible. To prevent loa¬

ding effects, a buffer stage E., - consisting of an emitter follower - is used. The

output of the VSSG is finally eg.The circuit diagram of the VSSG is given in Fig. (3.1.2). This is not the

complete circuit diagram. The circuits missing are those that give e„ ... e. from

s9 *'' sll rcsP60*176^ refer back to Fig. (3.1.1). These circuits have been omit¬

ted because they are exactly the same as the circuits that give e. from x, and

which have been given in Fig. (3.1.2).

The question as to how the various circuits work, that is for example, how

does the astable or the monostable multivibrators work, will not be answered here,

because such circuits have been widely covered in the literature on pulse circuits

[ 22,26]. fi will only be mentioned that with regard to the astable multivibrator used,

the diodes D. „ and the resistors R, in Fig. (3.1.2) - and not those connected to the

collectors of T- and T„ - have been introduced to improve the decay-time of the

square wave output x-. In case of the simplest astable multivibrator, where instead

of the diodes a short-circuit is present and instead of the resistances referred to an

open circuit, on switching a transistor off, its collector voltage decays exponential¬

ly and relatively slowly towards its negative value. The decay is slow due to the

load connected to the collector of the switched off transistor and which consists

of the condenser connected to the base of the other transistor in series with an equi¬

valent resistance. The introduction of the diodes and the resistances essentially dis¬

connects the condenser from the collector of the switched off transistor and thus

speeds the drop of the collector voltage to its negative value.

Oscillograms representing an arbitrary selection of the signals e., x2, s*„ ...

s-g are given in Fig. (3.1.3) "a" ..."f" respectively. To show that eg may assume

a very large number of possible forms, samples of some e„ signals are given in

Fig. (3.1.3), "g" and "h", where the durations of T and h-...

h. are as shown

in Fig. (3.1.3) "a" ... "f".

- 43 -

Fig. (3.1.2) Circuit Diagram of the Variable-Steps Signal Generator

Leer - Vide - Empty

- 45 -

*1

»2

*S

R4

*5

*8

*7

*B

"9

R10

*!!

R12

^3

l6,7,

1 kfi

12 kfl

1 k Q potentiometer

47 kfl

100 kfl

520 kfl

1.2 kfl

10 kD

100 k O potentiometer

3.3 kQ

33 kO

2.2 kQ

47 n

4,5,10,12,13

8,9,11,15

D

14

1,2

OC76

OC71

OC72

OC80

OA85

OA5

'1.2

'3,7

3.2 u Ffor e-

having a frequency

of approximately

15 Hz

500 pF

0.03 uF

0.47 u F (by varying

it, the delay of the

monostable multivibrator

is varied)

50 pF

I Astable multivibrator and emitter-follower

II Inverter and emitter-follower

HI Differentiator and clipper

IV Monostable multivibrator and emitter-follower

V Inverter and emitter-follower

VI Inverter and emitter-follower

VII "Or" circuit and emitter-follower

- 46 -

(a) (e)

(b) (0

-««**!< tut

(c) (g)

(d) (h)

Fig. (3.1.3) Oscillograms of some of the VSSG Signals

a e6(t) e slg(t)b x2(t) f s16(t)c s13(t) g s1?(t)d s14(t) h s18(t)

- 47 -

The advantages of the signal generator built by the author are that it is com¬

pact, easy to use and is capable of providing an extremely large number of diffe¬

rent test signals- Four stages of monostable multivibrators have been used, ho¬

wever if required, more stages may be readily added.

3.2.0 The Variable-Fequency On-Off Signal Generator

In studying the effect of input modification on the response of underdamped

systems, it was felt that if the modifier consisted simply of a sampler that opened

and closed for a finite interval of time after the application of the input step, then

it should be possible by choosing the sampling duration and frequency to obtain

an improved system response. The reason why this was suspected will be mentioned

in a later section. The signal generator to be described was found suitable for in¬

vestigating the truth of the above statement. This generator will be called the va¬

riable-frequency on-off signal generator and will be simply referred to as the

VFOOSG. The signal-flow diagram of the VFOOSG is given in Fig. (3.2.1). A

Schmitt trigger transforms a sine wave e. into a square wave s-. The square wave

is then differentiated and clipped in B to give the train of pulses s,. The latter pul¬

ses close a transistor switch having a condenser connected across its terminals and

which is being fed from a current source, thus giving rise to the saw-tooth signal

s,. To s, is superimposed a dc voltage signal e„ and a relatively high frequency

signal e,. The sum is shown as a signal s.. This signal is then clipped giving s,-

Feeding s- into an inverter followed by a pulse-shaper, the final output e. is ob¬

tained. The complete circuit diagram of the VFOOSG is shown in Fig. (3.2.2). A

short description of how this circuit functions follows.

A sine wave e, of amplitude approximately six volts is applied to the Schmitt

trigger denoted by I. The output square wave Sj is then differentiated by means

of the network RjC,, giving a train of positive and negative pulses a*tof a very

short duration superimposed on a dc level of approximately -10 volts. The train of

positive pulses s,. is then applied to the base of an n-p-n transistor T,, which

clipps off the negative pulses resulting in s„ at the emitter of T,. s, is composed

of a train of positive pulses of amplitude approximately 8 volts superimposed onto

a voltage level close to -10 volts.

- 48 -

0=^-0 H3f

LiiuI

:%-

, 1) The Signal-Flow Diagram of the Variable FrequencyOn-Off Signal Generator

A Schmitt trigger

B Differentiator and clipper

C Saw-tooth generator

D A nonlinear element

E A pulse shaper

e. Input sine wave

Sj Output of the Schmitt trigger

s? A train of pulses obtained from s1

through differentiation and clipping

s„ A saw-tooth voltage obtained by dis¬

charging a condenser in C by means

of the train of pulses s„

s. The sum of the signals e„, e„ and s„

s_ The output of the nonlinear element D

when the input is s,

e4 The output of the VFOOSG

e„ A dc voltage

e, A relatively high frequency sine wave

T1 Period of the input sine wave e-

T„ The duration of the relatively high fre¬

quency oscillations per period T1

T3 The time the oscillations start with

respect to the beginning of the input

sine wave

- 49 -

From the characteristics of p-n-p transistors it may be seen that the collec¬

tor current, for a fixed base current, is largely independent of the collector volta¬

ge except for quite low voltages. Therefore, a capacitor charging from a supply

voltage through a p-n-p transistor, for example an OC71 transistor, will charge up

at approximately constant current. This fact was made use of in designing the saw¬

tooth generator denoted by HI in Fig. (3.2.2). The capacitor C„ is charged from

the battery voltage through the transistor T- whose base is fed from a constant

current source. The transistor T. shunts the capacitor C,. It also acts as a switch

which closes whenever one of the pulses of s, occurs. Consequently, the voltage sig¬

nal s, - that results is a saw-tooth one. The slope of this saw-tooth voltage signal

may be varied by varying the bias of the transistor T_. To prevent loading, a buffer

stage composed of an emitter-follower is used. Its output is s„. Using the three

terminal resistance network shown in the circuit diagram, the voltage signals s3,

e, and e„ are added and the sum is fed into a class B amplifier. The output vol¬

tage signal that results is reamplified and shaped into a square waveform by means

of a Schmitt trigger. The output of the Schmitt trigger is introduced to the base of

an emitter-follower to give the final output e..

Oscillograms of e-f s.., s,, s„, s5 and e. are shown in Fig. (3.2.3) "a...

f".

In Fig. (3.2.3) "g", an oscillogram is given which represents s5 when e3 is zero.

The oscillogram in "h" shows the voltage signal e4 for the latter situation. By vary¬

ing the magnitude of the sine wave e„ and/or the slope of the saw-tooth signal s3,the duration of the oscillations Tg may be varied. The location of the oscillations

with respect to the beginning of the input sine wave e., that is T„, may be varied

by varying the level of the dc signal e,.

- 50 -

- 51 -

*1 10kn ^3 100 kfl

*2 680 SI "l4 2.2 k fl

*3 Ik SI Cl 0.05 uF

R4 470 A C2 0.47 u F

R5 47 il Tl,2, 5,6,7,8,9,10,11 OC71

"b 100 H T3,4 OC140

*7 220 SI el Input sine wave having an amplitude

^ 3.3 k.fl of approximately 6 volts

*9 1 k n potentiometer e2 A dc voltage

*!<) 33 kXI. e3 A relatively high frequency sine wave

*!! 4.7 kSl e4 The output of the VFOOSG

R12 15 k II

I Schmitt trigger

n Differentiator and clipper

in Saw-tooth generator

rv Summing network and class B amplifier

V Pulse shaper composed of an inverter and a Schmitt trigger

VI A buffer stage composed of an emitter-follower

52

7"S

t++«

r^UUiU<IWffHiiMt*«l<Wt++«

(a) (e)

5 volts

m±

(b)

j* J i * * * * ,4^^^ mtffmu^ g t++++V

-**$-(f)

1-fi-T -H

(c) (g)

(d) (h)

Fig. (3.2.3) Oscillograms of some VFOOSG Signals

a e^t) e s5(t)b Sl(t) f e4(t)c s2(t) g s5(t) for e3(t) = 0

d s3(t) h e4(t) for e3(t) = 0

- 53 -

Chapter IV

Dual Mode Control Systems

4.0.0 Introduction

The step response of an underdamped system has a short rise time and a

long settling time, while that of an overdamped system has both a long rise time

and settling time. A system which operates in two modes, initially as an under-

damped system and finally as an overdamped one can be made to have a step respon¬

se with improved characteristics, that is, one with a short rise time as well as

settling time. Critical in the design of such systems is the instant of switching from

one mode to the other.

hi the case of sampled-data control systems, improved step response may

also be achieved if samplers having two possible pulse-widths h- and h, are used

and which could be switched from one pulse-width to the other on requested.

In this chapter the above mentioned methods are introduced and analog com¬

puter results are presented.

4.1.0 Theory

A second-order system such as for example the one having the transfer func¬

tion

1 s (s + a)

where K and "a" are constants, if placed in a closed loop, can be made to give an

oscillatory, critically damped or overdamped closed loop system step response by

simply varying the constant K.

If a dual mode amplifier is used and if switching from one amplification K1 to

the other K_ is possible, Fig. (4.1.1), then by choosing Kj large and K_ small and

starting with K, and switching later to K„, the closed loop system can be made ini¬

tially underdamped and finally overdamped. It is worth noting that the amplifica¬

tions K- and K», the instants of switching from one value to the other and the loca¬

tion of the dual mode amplifier within the closed loop system effect the system

- 54 -

response. The first two points are verified in section (4.3.0). To show the truth

of the third point, suppose that instead of a single dual mode amplifier two such

amplifiers are used with the first capable of providing amplifications K«, K- and

the second K., K„ where

K, = *SK4 and K„ K5 K6 (4.1.2)

If the pair of dual mode amplifiers is placed in cascade and in the position previous¬

ly held by the single dual mode amplifier, the operation of the system will remain

unchanged. If, however, one of the two dual mode amplifiers is seperated from the

other, as for example is shown in Fig. (4.1.2), then the system response using

the single dual mode amplifier of amplification K1 and K„ will be different from

that using the pair of dual mode amplifiers having amplifications K_, K. and K.,

r,^Q'e(t) B)

K*

me^t).

Fig. (4.1.1) A Feedback Control System having a Dual Mode Amplifier

r (t) input

e (t) error

c (t) response

e1(t) input to G,(s)K

G« (s) a transfer function. G, (s) = i \1 1 sis + a.)

where "K" and "a" are constants. K = 1.

K-g amplifiers having amplifications K.. and K,

C computer

(1)(2) possible positions of the switches

- 55 -

c nJ—r__

_._= -.

I1

| K3 ,II | K4J. J.

<n j"" zrc(t)'

A| J(2) I"

1 J 1

^yUl s+a(2) t7\

s

K5 K6

Fig. (4.1.2) A Feedback Control System employing a pair of Dual Mode ,

Amplifiers

K„ c a dual mode amplifier with amplifications K„ and KgK.

„ a dual mode amplifier with amplifications K. and K»

Kg respectively. To show this, consider the differential equation of the closed loop

system

^L+ a, Wo #*-«:•*) = u; r(t)at

(4.1.3)

where £ is the system relative damping and oo. its natural frequency. ForG|(s)given by equation (4.1.1),

\V^T

and coo -f%l (4.1.4)

Taking the Laplace transform of both sides of equation (4.1.3), the following is ob¬

tained

s2C(s) - sc(O)- gp^+ 2 f uo [sC(s) - c(O)] - U^C(s) = co^R(s)(4.1.5)

which on rearranging gives

- 56 -

R(s) + sc(0) +|^2L + 2^oo c(O)C(s) =

5 5—2 (4-1.6)

s +2$ Wos+wo

or

C(s) = f [R(s) s, coo, c(0) §&] (4.1.7)

If equations similar to equation (4.1.6) are written for the second mode of

operation of the systems given in Figs. (4.1.1) and (4.1.2), and if the switching

instants are taken to be the same, then it will be seen that the equations are identi¬

cal except for the.,

' ' term which is the reason why if dual mode amplifiers are

used, their location within the loop is critical. This result is to be expected since

a dual mode amplifier is a nonlinear amplifier.

In finite pulse-width sampled-data control systems, if the sampling frequency

is several times the dominant frequency of the system, then sampling may be consi¬

dered to be amplification with an amplification factor less than one. This is also a

consequence of the theory developed in chapter H. The sampler may be replaced

by an amplifier having an amplification of one when h = T andwhen h = 0.5 T an ampli¬

fication of 0. 5, that is an attenuator of value 0.5 Thus if

hl Klhf

=

K^ ("-I)

see Figs. (4.1.1) and (4.1.3), then the change in the relative damping for both sy¬

stems referred to will be the same. This is the basis of the method used to improve

the sampled-data system step response as shown in section (4.3.0).

To obtain an improved transient response using a dual mode amplifier (or dual

mode sampler), the amplifier (or sampler), need not be placed in the forward path of

the closed loop system, but can be also placed in its feedback path. If such is the

case, then the system relative damping and natural frequency are given as follows

reCompare the above equation with equation (4.1.4). In the following section the analog

computer circuits are given which were used to obtain the results presented in sec¬

tion (4.3.0).

- 57 -

I H"_L> Ie(t) h,^Wlr,;u^Jr4TsiiiiJ

c(t)—>•

Fig. (4.1.3) A Sampled-Data Control System having a Dual Mode Sampler

e (t) the sampled error

Ph- „ the pulse widths of the dual mode sampler

4.2.0 Analog Computer Circuits

The systems of Figs. (4.1.1) and (4.1.2) were simulated using the analog com¬

puter circuit shown in Fig. (4.2.1). When s.(t)-c(t) is positive, comparator MOJ

is in its (+) state and connects the output of potentiometer Q02 to the input of P03.

When s.,(t)-c(t) is negative, the output of Q02 is connected to the input of Q03. The

switching of comparator MIJ from one state to the other is also governed by a si¬

milar condition. When s.,(t) is equal to s,(t), both comparators switch at approxima¬

tely the same time. This is needed for the simulation of the system of Fig. (4.1.2).

To simulate the system of Fig. (4.1.1), it is required that comparator MIJ remains

in one state. Choosing the latter state to be the (+) state, this may be achieved by

adjusting the coefficient of Pll so that s2(t)-c(t) remains always positive. If this is

not possible using the circuit of Fig. (4.2.1), then by connecting MIJ to ground in¬

stead of to the output of integrator A01, this difficulty will be overcome.

The circuit of Fig. (4.2.1) has the advantage that the quality of the system

response is independent of the magnitude of the input step.

As a simulation of a sampled-data control system having a dual mode sampler,

the circuit in Fig. (4.2.2) was used. Referred to by I is the closed loop system. With

switches "12", "13", "21" and "20" in position C, the system is a continuous one.

When switches "12" and "13" are in position L and switches "20" and "21" in posi¬

tion C, the system becomes a sampled-data one with the sampler in the forward

path. Letting switches "12" and "13" take position C and switches "20" and "21"

- 58 -

Fig. (4.2.1) Analog Computer Circuit used to simulate the Control Systems of

Figs. (4.1.1) and (4.1.2)

r(t) the input and is a step signal

c (t) the output

c (t) the derivative of the output

s- „(t) the signals which together with -c (t) determine the state

' of comparators MOJ and MIJ respectively

- 59 -

position R, the sampler is then located in the feedback path, n is a sine wave ge¬

nerator. HI is a circuit for generating a sinusoidal signal having the same frequen¬

cy and amplitude as that generated by n, but of a different phase which can be varied

as requested. To obtain the latter sine wave, the following identity was used

sin [kt + m(t)] = sin kt cos m(t) + cos kt sin m(t) (4.2.1)

When switch "03" is in position L, m(t) can have two values depending upon

the state of comparator M2J. The latter state in turn depends upon whether s (t) -

given in IV - is positive or negative.

With the comparator connection shown in Fig. (4.2.1), the sampling frequency

cj is related to the frequency "k" of the sinusoid generated by II in the following

way

U = 2k (4.2.2)

4.3.0 Results

With the switches of Fig. (4.1.1) in position (1), K, was adjusted to give the

step response shown in Fig. (4.3.1), (a), and which has a relative damping of 0.2.

The derivative of this response is given in (b).

With the switches in position (2), K, was selected to give ^ = 2.0. Operating

the switches so that initially they are in position (1) and when c(t) exceeds 0.065r(t)

are switched to position (2), the step response and its derivative shown in Fig. (4.3.1),

(c) and (d) are obtained. The settling time - defined here to be the time required for

the system response to settle to within 5 % of its steady state value - is for the

response shown in (a) 6 units and for that shown in (c) where the dual mode ampli¬

fier was used is only 3.75 units, where the unit of time is chosen to be the length

of a square.

Curves which are compared in this dissertation have the same time scale un¬

less otherwise stated. Further, the zero level in case of curves recorded from the

analog computer is located at the center as shown in Fig. (4.3.1).

Using the dual mode amplifiers as shown in Fig. (4.1.2) with K-, K., K-

and K. satisfying equation (4.1.2) and

(4.1.3)

Fig.

of

System

Control

Sampled-Data

the

simulating

Circuit

Computer

Analog

(4.2.2)

Fig.

o

- 61 -

(1)(2) (1): From ADO

(2): From A02

I closed loop system and sampler

r (t) input and is a step

e (t) the error

c (t) the output

c (t) the derivative of the output with respect to time

n a sine wave generator

k frequency of sinusoid generated by II and is equal to ten times the value

of the coefficient of Q01

HI circuit for generating sin [kt + m(t)]

m (t) output of potentiometer P19. m (t) = h where "h" is the

sampler pulse width

IV circuit for controlling the state of comparator M2J

s (t) output of amplifier A27 and is the signal that determines

the state of comparator M2J

K3 K4

K5 K6

and further with switching from one state to the other occurring when c (t) exceeds

0.83r (t), the results in Fig. (4.3.2) were obtained. When ^ was equal to 0.2 the sett¬

ling time was 12 units, Fig. (4.3.2), (b), while when dual mode amplifiers were used

as described above, the settling time was reduced to 3.4 units, Fig. (4.3.2), (d). The

settling time depends upon the initial and final relative dampings of the system and

upon the switching instant. To obtain satisfactory results using a single dual mode

amplifier, Fig. (4.1.1), the switching instants should be close to the values presen¬

ted in Fig. (4.3.3) and which are plotted in Fig. (4.3.4).

When two dual mode amplifiers are used as is shown in Fig. (4.1.2) and if the

dual mode amplifiers satisfy equation (4.3.1), then the recommended switching instants

are those given in Fig. (4.3. 5).

hi case of sampled-data control systems, system response can be improved if

dual mode samplers are used. Analog computer results illustrating this are shown

in Fig. (4.3.6). (a) and (b) represent -e (t) andc(t) respectively, see Fig. (4.2.2),

where h is approximately equal to T. The settling time exceeds 24 units.

- 62 -

Fig. (4.3.1) Improvement of System Response using a Dual Mode Amplifier

- 63 -

(a) step response of underdamped second-order system, ^ = 0.2

(b) -c (t), the negative of the derivative of the step response given in (a)

(c)l c (t), the step response of the second-order system when a dual mode amp¬

lifier is used as in Fig. (4.1.1) Initial relative damping f -= 0.2 and

final relative damping f „= 2.0.

(d) -c (t), the negative of the derivative of the step response given in (c)

To obtain the results given in (a) ... (d), the analog computer circuit of Fig. (4.2.1)

is used.

To obtain curves (c) and (d):-To obtain (a) and (b):

00 C

10 R

11 L

P02 0.5

Q02 1.0

POO 1.0

P01 1.0

P03 0.324

Q03 0.0324

P04 1.0

Q04 arbitrary

Q00 0.72

P10 0.99

Pll 0.99

00 C

10 R

11 C

P02 0.5

Q02,P00,P01 1

P03 0.324

Q03 0.0324

Q00 0.72

P04 1.0

Q04 arbitrary

P10 0.0325

Pll 0.99

1.0

The scale for (a) and (c) is the same and for (b) and (d) also the same.

- 64 -

—— -

r—

1— —

I--

-• —

j

_E

— —

—

—

/ -

/ i

/00

-

— ,-

-- Id— ;- --1

-

r _-

-

-=- ._= - -- -4- — -

-"^- - -"

'"

= ^ i= r^rrj

(a)

(b)

— ----

—

: z.-

_- - _-— - - ^ =

—

=r'~'

= /^ —;== .— -^ — r_. =: '3 = =[~

—-~—

—

- —

=-_

—

=

t= —

.— _:-J

—. HE:-

=z Z= ^

—

-- -H—

-=: — -

—

- — —~-

-

.r —\ = "="

--

-=: —i

--

—'= =

—

=—— h; ^_ — —

-=.= = ;~ ~-:

~

~- ~- Z^L

—:"

"_~

=i -. Z- =" '—:—

— — —

— - — —

—-= - - == -- =r

—

— —

— _

-j—

~ —

~."

_— "__ ^a

~

- ^-~

^ "^ = =— ^ ~i= == ^_

—

-- -f - —

~

_ _

~ —

— —

- 3--—"

t=_ — — — -

_.

- —

_= = ^- -=-

ZZLZ—

-— :===— - = •=-

=:— —

=. r= —.

=—

^---

= ^ -- =

(c)

(d)

Fig. (4.3.2) Improvement of System Step Response using a Pair of Dual Mode

Amplifiers

- 65 -

(b)(a) The step response and its derivative of an underdamped second-order sy¬

stem having £ = 0.2

(d)(c) the step response and its derivative of the system used in (b) and (a), when

a pair of dual mode amplifiers is used as shown in Fig. (4.1.2). f -= 0.2

and ^ „= 2. 0. Further, the equations (4.1.2) and (4.3.1) are satisfied.

To obtain the results given in (a) ... (d), the analog computer circuit of Fig. (4.2.1)

is used.

To obtain (a) and (b):- To obtain curves (c) and (d):-

00 C

10 R

11 L

P02 0.5

Q02, POO, P01

P03 0.324

Q00 0.72

P04 1.0

Q04 arbitrary

P10 0.99

Pll 0.99

0.5 Q02.P00.P01 0.5

00 C

10 C

11 C

P02 0. 5

Q02,P00,P01

P03 0. 324

Q03 0. 0324

P04 1. 0

Q04 1. 0

Pll, P100. 0325

Q00 0. 72

Scale in (b) and (d) the same and scale in (c) five times that in (a).

- 66 -

<N^ 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.1 0 0 0 0 0.01 0.01 0.01

0.2 0.03 0.07 0.07 0.07 0.08 0.08 0.08

0.3 0.10 0.14 0.16 0.16 0.17 0.17 0.18

0.4 0.16 0.26 0.28 0.29 0.30 0.30 0.31

0.5 0.28 0.38 0.41 0.42 0.43 0.44 0.45

0.6 0.45 0.58 0.60 0.61 0.62 0.63 0.63

0.7 0.70 0.75 0.78 0.80 0.81 0.81 0.82

0.8 0.90 0.92 0.93 0.94 0.95 0.96 0.97

Fig. (4.3.3) Recommended Switching Instants when a Single Dual Mode Amplifieris used as in Fig. (4.1.1)

SiS2

initial relative damping of second-order system

final relative damping of second-order system

are the values tabulated above and are given by

c(tjK_ TW ,

where r(t) is a step input. Thus

switching from \ - to ^ ,should be performed

when c(t) exceeds K r(t).

- 67 -

Fig. (4.3.4) The Recommended Switching Surface when a Single Dual Mode

Amplifier is used to improve System Step Response

*1

<2

K

initial relative damping of second-order system

final relative damping of second-order system

the values given In the table. When c(t) exceeds K r(t) swit-s

ching from ? < to ^ ,should be performed in order to obtain a

a satisfactory response

XJ21.0 1.5 2.0 2.5 3.0 3.5 4.0

0.1 0.58 0.60 0.61 0.61 0.62 0.63 0.65

0.2 0.83 0.83 0.84 0.85 0.87 0.88 0.89

0.3 0.90 0.91 0.91 0.92 0.92 0.92 0.93

0.4 0.92 0.94 0.95 0.95 0.96 0.96 0.97

0.5 0.99 0.99 0.99 1.0 1.0 1.0 1.0

Fig. (4.3.5) Recommended Switching Instants when a Pair of Dual Mode

Amplifiers is used

f initial relative damping of second-order system

£ final relative damping of second-order system*2

- 68 -

s

=

=

s£==

=

== ^=.

=

^ =

=1

X

1 1H1=

I1

1 oo= 1

=

s

j|ln

(a)

Ills (b)

(c)

(d)

Fig. (4.3.6) Improvement of System Response using a Dual Mode Sampler in

the Forward Path

-

- 69 -

(a) -e (t), the negative of the sampled error when h = T

(b) c(t), the system step response when h = T

(c) -e (t), when ht = T and h2 = (0.01)T

(d) c (t) corresponding to the latter case. Switching from h. to h, occurs

when c (t) exceeds 0.24 r (t)

To obtain the curves in (a) ... (d), the analog computer circuit of Fig. (4.2.2) was

used.

To obtain the curves (a) and (b):- To obtain curves (c) and (de¬

position of switches and potentiometer

settings as above except:

10 C

12 L

13 L

20 C

21 C

03 C

02 L

01 R

P05 0.5

Q05,Q06,Q07

Q01 0.1

P06 0.324

P07 0.72

P19 0.9

0.05

03 L

P24 0.9

Q17 0.12

P19 1.0

Q30 0.009

Recording speed: 0.8 mm/sec, where the length of a square corresponds to 5 mm.

- 70 -

When a dual mode sampler is used having h, = T and h„ = (0. 01) T with switching

from h1 to h„ occurring when c(t) exceeds 0.24 r(t), a step response having a

settling time of 9 units is obtained, Fig. (4.3.6), (d). In (c), the corresponding

curve of -e (t) is given.

When the dual mode sampler is placed in the feedback path improved res¬

ponse is also possible. Adjusting the amplifier in the forward path of the feedback

system so that when h = T the system response is overdamped - J = 2.0 - as

shown in Fig. (4.3.7), (b) then operating the dual mode sampler so that h.. = (0.01) T

and h, = T with switching from hj to h, occurring when c(t) exceeds 0.96 r(t), the

response in Fig. (4.3. 7), (h), is obtained. This response has a settling time of

3. 5 units as compared to 10. 5 units for that given in (b). In (a), -c (t) is shown

for the case where h = T, and the same signal when the dual mode sampler is used

is given in (g). Shown in (c) and (d) are sin kt and sin [kt + m(t) 1 respectively. The

effect of switching from a phase m. - which gives h.. = (0.01) T - to a phase m, -

which corresponds to h, = T - is clearly shown in the latter waveform. In (e), c(t)

is given and is seen to be constant over an appreciable interval of time. The reason

is, that during this interval the system is nearly open loop and because G.,(s) is of

the form given in equation (4.1.1), the response is approximately the integral of

the input step. In (d), s(t) is shown.

From the results presented in this section, it may be concluded that dual mode

amplifiers and samplers can be used to improve the step response of control systems.

- 71 -

-cJO (a)

0>)

sinkt

sin kt+m(t)

(c)

(d)

Fig. (4.3.7) (a,b, c,d) Improvement of System Step Response by using a Dual

Mode Sampler in the Feedback Path

- 72 -

c(t)

s(t)

-CpW

c(t)

(e)

d)

m fe)

(h)

Fig. (4.3.7) (e,f,g,h) Improvement of System Step Response by using a Dual

Mode Sampler in the Feedback Path

- 73 -

(a) -c (t), the negative of the sampled response, refer to Fig. (4.2. 2). h = T

(b) c(t), the system step response. $ = 2.0

(c) sin kt

(d) sin [kt + m(t) ] nij= 0.9, m2

= 0.009, k = 1 rad./sec. Switching from

m. to m, occurs when c(t) exceeds 0.98 r(t)

(e) c(t), the derivative of the response

(f) s(t)

(g) -cp(t)(h) c(t)

To obtain curves (a) ... (h), the analog computer circuit of Fig. (4. 2.2) was used.

To obtain curves (a) and (b):- To obtain curves (c) to (h), same as

in (a) and (b) except:-10 C

12 C

13 C

20 R

21 R

03 C

01 R

02 L

P05 0.5

Q05,Q06,Q07

Q00 0.5

P19 0.9

P06 0.324

P07 0.72

Q01 0.1

1.0

03 L

P24 0.009

Q17 0.48

P19 1.0

Q30 0.9

Recording speed: 0.8 mm/sec., where the length of a square corresponds to 5 mm.

- 74 -

Chapter V

Nonlinear and Signal Adaptive Control

5.0.0 Introduction

In the previous chapter it was shown that dual mode amplifiers and samplers

can be used to improve the step response of control systems. In this chapter non¬

linear amplifiers and signal adaptive samplers are used to improve the step respon¬

se of control systems.

Nearly a decade ago nonlinear amplifiers were employed to improve the step

response of feedback control systems [34]. Lewis [21] studied a second-order

system having a relative damping of the form

f = Kj [K2 - Kg |r(t)l + K4 c(t)] (5.0.1)

where K., K„, K„ and K, are constants, r(t) the input and c(t) the system output.

Caldwell [4], in his doctoral dissertation, developed the method suggested

by Lewis and compared it with optimum relay servo using a differential analyzer.

In this chapter, an analog computer study of a second-order system is presen¬

ted in which the system relative damping is varied as a function of signals present

in the control system. A comparison of the responses is made and further it is

shown, that nonlinear amplifiers may be used to improve the step response of hig¬

her-order systems. Analog computer results are given for a case where a third-

order system was used.

In chapters II and IV was shown that sampling and amplification are in many

ways similar. Consequently it is to be expected that if nonlinear amplifiers impro¬

ve the step response of control systems, then the same should be true for nonlinear

samplers or signal adaptive samplers as is usually referred to the latter in the li¬

terature.

Billawala [31 - in his master's thesis - verified this experimentally for a

sampler located in the path of the error where

h = f [le(t)l] (5.0.2)

- 75 -

Hamza [13] employed a sampler whose pulse-width was a function of the square

of the error derivative and which was also located in the system forward path to

improve the step response of a second-order system.

Li both papers referred to the functional relationships were limited, that is

limiting was used.

The author of this dissertation studied signal adaptive sampling for a large

class of functional relationships and in section (5.3.0) a sample of the results ob¬

tained is presented.

5.1.0 Theory

The differential equation of a second-order system may be written as

dt

where \ is the system relative damping, CO its natural frequency, r(t) its input

and c (t) its output.

In chapter IV it was shown that if a second-order system had initially a small

relative damping and if at an appropriate time after the application of an input step

its relative damping was suddenly increased, then the system step response can be

made to have a relatively short settling time. If the transition from the initially

small relative damping to the finally large one does not occur abruptly but instead

rather gradually, improved step response may be obtained. Referring to Fig. (5.1.1)

this may be achieved if

K = A + f [e(t)]n

(5.1.2)

where "A" and "n" are positive constants.

Improved step response is also to be expected if initially the system relative

damping is large and soon after the application of the input step it is decreased to

accelerate the system and finally it is increased to decelerate it. Such a variation

in the relative damping of the system may be obtained if "K" satisfies the following

equation

76

K = A + f (x) (5.1.3)

where "A" is a constant and "x" is given by equation (5.1.4).

C

Y\V

K ^(s)Uem

G^s)

Fig. (5.1.1) System and Computer Controlled Sampler

Transfer function of second-order system

K Nonlinear amplifier

C Computer

r (t), e (t), c (t) The input, the error and the output respectively

dne (t)

dtn

dne (t)dtu

d]c(t)

dt

(5.1.4)

where n, m, j and k are positive constants.

Relationships similar to the latter exist when a signal adaptive sampler is

used. These relationships are obtained by merely replacing "K" in equations (5.1.2),

(5.1.3) and (5.1.4) by "h" or by "T" ", where "h" is the sampler pulse-width and

"T" the sampling period. Obtaining analytical solutions for the system step respon¬

se when the methods of control described above are used is for the large majority

of cases not possible due to the presence of nonlinearlties and even for the very simp¬

lest cases requires a great amount of labour, so that the best procedure to design

such control systems is to resort to analog computer simulation.

- 77 -

5. 2.0 Analog Computer Circuits

Presented in this section are the analog computer circuits which were used

in obtaining the results given in section (5. 3.0). The computer employed was the

PACE 231R analog computer which is at the computation center of the Institute for

Automatic Control and Industrial Electronics of the Swiss Federal Institute of Tech¬

nology.

hi Fig. (5.2.1) the analog computer circuit simulating the control system of

Fig. (5.1.1) is shown. With switches "20" and "21" in position C the system is a

second-order one and when they are in position L, a third-order system is obtained.

The nonlinear amplifier, Fig. (5.1.1), is simulated by the multiplier M1G. The con¬

stant "A",refer to equations (5.1.2) and (5.1.3), corresponds to the output of po¬

tentiometer P12. The inverter A02 and the switch "23" are used to assure that f.(t)0 1

be positive. The output of A00 is equal to (coetf-e-e*nt 0f Aqq ) times its input.

The circuit consisting of AB26, AB27, A32 and A13 is used to give a signal

f5(t) having a negative sign. This is necessary in order that the sign of eAt) be the

same as that of -e (t).

To study sampled-data control systems having signal adaptive samplers, the

circuits in Fig. (5. 2. 2) were used. Circuit I is a simulation of a second-order sy¬

stem. By adjusting switches "12", "13", "20" and "21" a continuous or sampled-da¬

ta control system may be obtained. The location of the sampler can be chosen to be

in the forward and/or feedback path of the system. Circuit IT is a sine wave genera¬

tor which gives sin [$ (t) t] as an output. When switch "01" is in position R, $(t) is

constant and is equal to ten times the coefficient of Q01. m is a circuit for genera-

ting-sin [ $ (t) t + m (t)] and IV to DC are special purpose circuits and will be descri¬

bed in the following section.

- 78 -

tire^g>^

Fig. (5. 2.1) Analog Computer Circuit simulating the Signal-Flow Diagram of

Fig. (5.1.1)

- 79 -

CO

,_|

o -M

hi 34-» aC +?O 3

(J co

O

Data urati CQCU

i

a•o £

cu CUi-H

'bo

amp •i >COhi

*-» i—i+J

ingS;cu

"p. (t)l ontrc circustudy ive

Sa UTSSI

+

ng

a

c

mplei

ut.

ed

in

1 f havir

exae

inpco

3-a

o

CQ c

cuCQ

vh^

CQ

6 « •r-l

CQ3

.o

CQ *\

3 CO a 1^1

d S.S

Cin iving yste rato: ating cing ircu ecuu CQ 0) hi 3 u A

cu Js h| e cu •5 cu*"•w

CU cu c o CQ O3 CQ T3 bo cu hi O**

i1 Ocu

bfi

hi hi &1aaa*

o

o

bfi

ts enri¬cu

itfo itfoO ch c 3 3 .3 o

•a cu •i-H O u o o,CO 10 hi hi cu o

•3 < «U O•l-t

££

n" acm' ^

inr-t a 0 £; >

.S>fa

Leer - Vide - Empty

- 81 -

5.3.0 Analog Computer Results

In Fig. (5. 3.1), (a), the step response of an underdamped second-order sy¬

stem is given. This response has a settling time of 3 units, In (b), the system

step response is shown when a nonlinear amplifier is used whose amplification "K"

is given by

K = A + B [e(t)]2

(5.3.1)

where A and B are constants. The settling time in (b) is 2 units. In (c), "K" is gi¬

ven by

K = A + B [c(t)] (5.3.2)

and in (d)

K = A + B [c(t)]2

(5.3.3)

The settling times for (c) and (d) are 3 units and 5 units respectively. In (e), the re¬

lationship is

K = A + B fe(t) • c(t)] (5.3.4)

and in (f) it is

K = A + B [e(t) . c(t)] (5.3.5)

The settling times for the latter two cases are 4 units and 2.5 units respectively.

In obtaining the curves given in Fig. (5.3.1), "A" was chosen such that for

K = A the relative damping of the system was 2.0 and B was varied until the system

response was the best attainable with respect to overshoot and settling time.

By using division as shown in Fig. (5.2.1), the system response may be

made independent of the magnitude of the input step. In the method suggested by

Lewis [21], the quality of the step response depended upon the magnitude of the

input step which is a disadvantage. Curves (g) and (h) show that if division is used the

quality of the step response becomes independent of the magnitude of the input step.

- 82

00

(c) (A

(e) (0

(i) 0)

- 83 -

=E= ==

-

—

=—

-

—

—

—

— —

—

-.= = :~ H—:

= =-

~

= -_=

— -- — — —

—

—" ^ =:—;

= ~--

"= 3

e]^= = - —

-=

(k)

Fig. (5.3.1)

(a) c(t)

(b)

(O

-

_

-.

- -

-

\

-J

-

'•,

III III

-

_

-

-

-=1 -_

c(t)

ep(t)

- 84 -

(d)

(e)

~

= -- --3 "3 *r- r- -" ~. -. '- -„" n- --'-

f_: r'-l ^ i-- -

.- i' -

.---_ _

- -.-. -_J - -- — .-- i.- =--

pi --- _J~, "_ -.

~ ~

w

vh if :-_

~- --~ — :--- ^ •- '-• ".- ~i- .-f .--•. ^ :1i -- ii :-: ->

~_ ^ .= iii ~ r.. .:.

—

--•' -: =

fe fi -- i~f ~ --. _- '-; -- -•; --

-

= =; ^£ 3 -

~ .V -

~

-; -.- v- -i .£ -.

.

— -: = = -^ ~H :"-" ii= - i; ~_ ii V -=

-~ — -- i; =1 2 ~zl :: ---~

_-r. 7= -•- fii

s =f> r« 5* 51 3r .i^ 7-.-

-; --. ~ p.„ 7 ;-'-' -:

m fl ft ft rii- T V - V-. -

•.--

= ^-_-•-

-;

-

-- N _ -1 -.-"

r= ^ = ^ =- -:.: i r" = -if --- -- = ~

s S = fi-r- 'P. ^ ^ = = ~] ii £- "iff ~[ -T:

f3=-

-~ = Si — = -.-_ ~ "- =-_ "-

c(t)

«Pw

ff) 11 -m(t)

fe)

(•'.;- .=-' = ;-

"H"

jj; :;. '-- -:-U

•- -- '-- -'-

vr =1 if. 'ii :r; = •

—- :-

-:- ^ -

•--

'- "- -": i- :. -'- ~ ~

"

--"

- - --

-- •:- -%; -.- i£; -;. :' if ii. ' if ':-: -- -'

c(t)

W c(t)

Fig. (5.3. 2) Improvement of System Step Response using a Signal Adaptive Sampler

- 85 -

a... h Second-order system step responses

i, ] Third-order system step responses

k f-(t), refer to Fig. (5.2.1), in case of response (j).

Referring to Fig. (5.2.1), data to obtain curves a ...k:-

(a)

(b)

(c)

(d)

(e)

&=conscant

11 C

00 L

20 C

21 C

K=A+B[e(t)]2

11 c

20 c

21 c

02 L

22 L

03 L

01 L

23 C

00 C

K=A+B[c(t)]

11 C

20 C

21 C

13 C

10 C

01 R

23 R

K=A+B[c(t)]2

same as in (c) except13 L

10 L

23 C

K=A+B[e(t) • c(t)l

same as in (d) except

02 R

13 R

12 L

01 C

23 R

Settling time:-

Q03 0.5

P10 0.324

Q10 0.72

P12 1.0

Settling time:-

Q03 0.5

P10 0.324

Q10 0.72

P12 0.01

P04 0.99

Q00 1.0

3 units

2 units

Settlinig time:-

Q03 0.5

P10 0.324

Q10 0.72

Q00 0.0803

P04 0.99

P12 0.1

3 units

Settling time:- 5 units

Q00 0.0022

Settling time:- 4 units

Q00 0.0242

- 86 -

(f) K=A+B[e(t) -c(t)]

same as in (e) except

02 R

12 R

01 C

23 R

fe) K=A+B[e(t)]2

same as in (b) except

Q00 0.1

Q03 0.5

(h) K=A+B[e(t)]2

same as in (g) except

Q03 0.25

(i) K=constant

11 C

20 L

21 L

00 L

(i) K=A+B[e(t)]

11 C

20 L

21 L

02 L

22 L

03 L

01 L

23 C

00 C

00 K=A+B[e(t)]2

same as in (j)

Settling time:- 2. 5 units

Q00 0.413

Settling time:- 2.2 units

Overshoot :- 50%

Settling time:- 2.2 units

Overshoot :- 50%

Settling time:- 6. 5 units

Q03 0.5

P10 0.324

Q10 0.72

P01 0.9

Settling time:- 2.3 units

Q03 0.5

Q10 0.72

P10 0.324

P01 0.9

P04 0.99

P12 0.1

Settling time:- 2 units

Referring to Fig. (5.2.2), the data to obtain curves a ...h is as follows:

(a) h=0.445T ; T=constant

Circuits used:- I, n, HI

10 C

12 L

13 L

20 C

21 C

02 R

01 R

P05

P06

P07

Q05,Q06,Q07Q13

Q10Q01

Settling time:- 11 units

0.5

0.324

0.72

0.25

0.4

0.5

0.4

- 87 -

(b),(c) h=B[e(t)]3

; ht=Q = 0.39T ; T=constant

Circuits used:- I, n, m, K Settling time:- 5.5 units

same as in (a) except

Q13 0.35

P24 0.125

(d)..(f) h=h1-B[c(t)]n

; h1=0.445T ; B=15.4T ; n=l ; T=constant

Circuits used:- I, n, m, VHI Settling time:- 7.25 units

same as in (a) except

22 C Q13 1.00

02 L P44 0.5

P42 0.065

(3) From A05

Output of A44 to input of A21

(g) h^j-Btc (t)]n ; h1=0.445T ; B=15.4T ; n=2 ; T=constant.

Circuits used:- I,n,m,Vin Settling time:- 6 units

same as in (g) except

P30 0.192

P31 0.76

Input to A30 from output of A07

Input to A31 from inverse of output of A05

Output of A33 to input of A21

- 88 -

Signal dependent amplifiers may also be used to improve the step response

of higher-order systems, hi (i), the step response of a third-order system is given

and in (j), the step response of the same system is shown when a signal adaptive

sampler is used whose amplification "K" is of the form

K = A + B [e(t)]2

(5.3.6)

The settling times for (i) and (j) are 6.5 units and 2.3 units respectively. In (k)

f_(t) for the latter case is shown.

For sampled-data control systems the following results were obtained:

In Fig. (5. 3.2), (a), the step response of a second-order sampled-data con¬

trol system is given. The sampler was located in the path of the error and had both

a constant pulse-width and sampling period with h = 0.445T.

Operating the sampler as a signal adaptive one having T constant and

h = B [e(t) ]n (5.3.7)

where "B" and "n" are positive constants, improved step response can be obtained.

In Fig. (5.3. 2), (b), the step response of the sampled-data system is shown when

T is constant and h is governed by equation (5.3.7) with n = 3 and h (0) = 0.39T.

In (c) the sampled error e (t) is given. The settling time in (a) is 11 units and in

(b) is 5. 5 units.

Operating the signal adaptive sampler according to relation (5.3.8)

h = hx - B [c(t»n

(5.3.8)

where "h..", "B" and "n" are positive constants, improved step response may be

also obtained. For n = 1, h- = 0.445T and B = 15.4T, the system step response

is as shown in Fig. (5.3.2), (d). In (e), e (t) is shown and in (f) [-m(t)] is given.

With n = 2, B = 1000T and "h" the same as in (d), the step response (g) resulted.

Its settling time is 7.4 units.

The pulse-width given as a function of two variables can also be used to impro¬

ve the step response of sampled-data systems. Li (h), the pulse-width was of the

form

h = Kx e(t) - K2 c(t) (5.3.9)

- 89 -

The settling time in (h) is 6 units.

Improved step response was further obtained by varying the sampling frequen¬

cy keeping the pulse-width constant. In Fig. (5.3.3), (a), the step response is shown

of a second-order sampled-data system having the sampler in the path of the error

and with both "h" and "T" constant. In (b) the output of A14 is given. The latter sig¬

nal controls the behaviour of the sampler. With the pulse-width the same as in (b)

but with $ (t) variable and given as follows

$(t) = Kj [e(t)]2

(5.3.10)

where Kj = 29.6, the system step response is as shown in Fig. (5.3.3), (c). Li

(d) is the corresponding output of A14 and in (e) the behaviour of sin [ $ (t) t] may

be seen. The settling times in (a) and in (c) are 11 and 5. 5 units respectively.

To obtain the results presented in Figs. (5.3.2) and (5.3.3) the signal adaptive

sampler was placed in the path of the error. Similar results may be achieved if the

sampler is placed in the feedback path, hi this case, however, the step response

of the continous system must be overdamped.

As an illustration, choosing ^ = 2.0, the system step response with the

sampler closed will be as shown in Fig. (5.3.4), (a). In (b) is the corresponding

[-e(t)] and in (c) is the system open loop step response. When a signal adaptive

sampler is placed in the system feedback path and with "T" constant and "h" given

by

h = hj - Kt Ve(t) (5.3.11)

where Kj = 1.273T and hj = T, the step response (d) is obtained. In (e), ' -e(t) '

is given and in (f) it is repeated using a faster recording speed to show clearly the

variation of the pulse-width, hi (c) the rise time is approximately 4 units and in (a)

and (d) the settling times are 14 and 5.3 units respectively which is a clear indica¬

tion of improvement in system step response.

In chapter IV dual mode control was used to improve the step response of con¬

trol systems and in chapter V nonlinear amplifiers and signal adaptive samplers

were used for the same purpose. In the following chapters other methods of control

are discussed which result in system step responses having shorter settling times

than those obtained in chapters IV and V.

- 90 -

(a) c(t)

(b) output of

A14

(c) c(t)

(d)

(e)

rr

I

output of

A14

sin $(t) t

Fig. (5.3.3) Improvement of System Step Response using a Signal Adaptive Sampler

- 91 -

Referring to Fig. (5.2.2), the data to obtain curves a ... e is as follows:-

21C(a),(b) h=constant ; Co =constant ; where Co =

-?=-

Circuits used:- i,n,iv Settling time:- 11 units

10 C Q00 0.5

12 R Q01 0.74

13 R P05 0.494

20 C P06 0.324

21 C P07 0.72

01 R Q05,Q06,Q07 0.25

al 0.8 P13 0.2980

bl -0.8

a2 0.85

b2 -0.2

s 0.6

b3 -0.6

Input to A13 from output of A02

(c)..(e) u>s=K[e(t)]2

; h=constant

Circuits used:- I, II, IV, V Settling time:- 5. 5 units

same as in (a) except

01 L

P22 0.4

Input to M0A from output of A07

Output of A23 to "L" of switch "01"

- 92 -

(a) c(t)

_J-

-

.

(b) - Ul1

J

1

-

-

-e(t)

(c) c(t)

(d)

(e)

-

'

--~

- r-~

•i 1 A" "

—

-

- -

_ .

-

^ - H -\

-j -

_

-

- —..

-

-

—

-:

-

-

i -

-

1 ' 1 " " "

milu*

-

c(t)

-e(t)

- 93 -

- -

-

- -

CJ.

-

m i!-""r f-r

i1 "

-

1 1 1 i; n *

* '

j.i

II

- -

-

_.

- - -

-

(f)

Fig. (5.3.4) Improvement of System Step Response using a Signal Adaptive

Sampler in the Feedback Path

(a),(b)

(c)

h=T=constant

Circuits used:-

10 C

12 C

13 C

20 R

21 R

02 R

01 R

h=0.01T ; T=constant

Circuit used:-

same as in (a) except

Q13 0.009

atato obtain curves a.... f is as follows:-

Settling time:- 14 units

i,n,m

Q00 0.5

Q01 0.2991

P05 0.5

Q05,Q06,Q07 0.5

P06 0.0032

P07 0.72

Q13 0.9

Settling time:- infinite

i,n,m Rise time :- 4 units

(d)..(f) h=hx - Kj ye(t) ; Hj=T ; K1=1.273T ; T=constant

Circuits used:- I, n, m, VI Settling time:- 5.3 units

same as in (c) except

02 L P40 0.5

P41 0.1273

Q13 1.0

Input to A40 from output of A08. Output of A41 to input of A21. Output of

P41 to an input of A41 having an amplification of 10. (+) of MOJ connected to

(-) of MIJ and vice versa. In (f) the recording speed is five times faster than

that in (e)

- 94 -

Chapter VI

Optimum and Quasi-Optimum Control. First-Order Systems

6.0.0 Introduction

In this chapter a method is presented for obtaining the optimum step response

to first-order open loop systems.

Consider the first-order system having the transfer function

G(s)s + a

(6.0.1)

where "s" is the Laplace transform variable and "a" is a constant. The pole of G(s)

in the complex frequency s-plane is located at s= -a. Since "a" is real, therefore

the pole of G(s) lies on the real axis and its location depends upon the value of "a".

The unit step response of G(s) is given by

c(t) = (1 e"at) (6. 0.2)

For "a" greater than zero, the pole of G (s) lies in the lefthalf of the complex fre¬

quency s-plane and the magnitude of c (t) approaches unity as time progresses. E

gets to be unity, however, only after infinite time has elapsed. By modifying the in¬

put step appropriately a much shorter response time may be obtained. The modifi¬

cation - which may be performed by means of a switching device, see Fig. (6.0.1)-

r(t)>-"I,

rm(t)Gfe) c(t)

Fig. (6.0.1) System and Computer-controlled Sampler

r(t) input

rm« modified input

c(t) output

m maximum input allowable

G(s) linear system

C computer

- 95 -

consists of applying to the system the maximum available torque at the instant the

step input appears and once the system output reaches the value required, the input

step is applied alone to the system, see Fig. (6.0.2).

6.1.0 Theoiy

It will be proved that if the step input to a first-order system is changed sud¬

denly to a value equal to the system output at that instant, then the system will

come to rest.

Proof: The differential equation which relates the output c (t) of a first-order

system to the applied input r (t) may be written as

£& + ac(t) = r(t) (6.1.1)

where "a" is a constant.

Taking the Laplace transform of both sides of equation (6.1.1), the following

equation results

s C(s) - c(0) + a C(s) = R(s) (6.1.2)

c(s) .

H(s) , c(0)(6A 3)

where c (0) is the output of the first-order system at t = 0.

- 96

rm(t)*c(t)

1

J-T -*t

Fig. (6.0.2) Modified Input and corresponding System Output

1 r (t), modified input

2 c(t), system output

For R(s) =£i-i

, equation (6.1.3) becomes

cw = c(0s;^(0)s

which may be written as

(6.1.4)

c(0)C(s) "

s(s + a)c(0)s + a

(6.1.5)

Taking the inverse Laplace transform of both sides of equation (6.1. 5) gives

at-,c(t) = c(0) [1 - e_al] + c(0) e

-atc(0) (6.1.6)

Q. E. D.

T-f see Fig. (6.0.2), may be expressed in terms of m, d and a, where

d = m-b, as follows:

COty = m [1 - e"aTl] = b

Therefore,

-aT,.

me 1 = m-b

or

--aT, d . . ,

e 1 = —,where d=m-b

m'

(6.1.7)

(6.1.8)

(6.1.9)

which finally gives

- 97 -

An alternate way of looking at this problem is to consider the controller that

modifies the input step to the first-order system, as acting on the input in such a

way so as to generate from it a real zero that cancels the real pole of the system.

Let Rm(s) be the Laplace transform of the modified input r (t). Rm(s) may

be expressed as

Rm(s) = | [m-de"sTl] (6.1.11)

see Fig. (6.0.2), where d=m-b. The zeros of equation (6.1.11) occur when equation

(6.1.12) is satisfied

e"sTl = =(6.1.12)

Expressing s in terms of a real and an imaginary component, that is, writing

s = ot + j p, therefore equation (6.1.12) may be written as

e(*+jp)Tl =d_

(m

or

e ocTl [cos (i Tj + jsin p Tj ] = ^ (6.1.14)

Equating the real and imaginary parts of equation (6.1.14), the following results

sin pTx = 0 (6.1.15)

e"Tl cos p Tj =i

(6.1.16)

From equation (6.1.15)

pTj =+ 2n IT where n = 0,1,2 ... (6.1.17)

Substituting equation (6.1.17) into equation (6.1.16) gives

- 98 -

eaTl =1- (6.1.18)m

or

^T,1"! (6'1-19)

From equation (6.1.17)

(j=+ ^- n = 0,1,2... (6.1.20)

The zeros of equation (6.1.11) are thus infinite in number and they fall on a line

parallel to the jcii-axis with each one seperated from the other by a distance of

211

T*

To cancel the real pole of the system, the zero of equation (6.1.11) which lies

on the real axis in the complex frequency s-plane, that is, the zero whose n = 0, must

fall on top of this real pole, or in other words, the following must be satisfied

oc=^ lni =-a (6.1.21)I- m

which gives

Tl —I^H f6-1-22*

Equations (6.1.22) and (6.1.10) are seen to be identical, which indicates that

the two methods used lead to the same result as was previously claimed.

If the pole of the first-order system lies in the right half of the complex fre¬

quency s-plane, similarly by designing a controller that generates a zero that can¬

cels this pole, an optimum response may be obtained. The controller in this case

should produce nonminimum phase zeros which, as may be readily verified, can be

obtained by first applying to the system a positive step then a large negative one.

In practice, however, such a controller will not be very practical, since any slight

disturbance or motion of the zeros will cause an instability. In contrast, if the pole

is in the lefthalf of the complex frequency s-plane and if the parameters of the sy¬

stem vary slowly, then the simple controller which has been previously described

should be sufficient to give an optimum response or in the worst case a quasi-op¬

timum one.

- 99 -

6.2.0 Experimental Study

To study this type of control in the laboratory, a first-order system was si¬

mulated using RC components and the modified inputs of interest were generated

using the VSSG which has been described in chapter m. Some of the results obtai¬

ned are shown in Fig. (6.2.1). Oscillograms (a), (b) and (d) show the effect of an

incorrect choice of T... The correct choice of T- is given in oscillogram (e). Oscil¬

logram (e) shows r (t) and c (t) for a case of optimum or quasi-optimum control,

depending upon whether the maximum amplitude of r (t) is the maximum amplitude

attainable or not. In (f), c(t) for the latter case is repeated and included is also

dc(t)dt

Li Fig. (6.2.2), oscillograms are given representing the behaviour of c(t) in

the phase plane. The oscillograms in Fig. (6.2.2) are "sort of" symmetrical about

the c (t) axis because the input signal r (t) used was actually a square wave with on¬

ly the positive steps modified as requested. This has the advantage that the response

to the step input and to the modified step input are obtained in one oscillogram, thus

making it convenient for the comparison of the two responses. The modifications per¬

formed were of the kind shown in Fig. (6.0.2). In Fig. (6. 2. 2-a) an arbitrary choice

of "m" and of "T." was made. By adjusting T.., in this case by decreasing it, oscil¬

logram (b) was obtained. The latter oscillogram represents a response of the type

shown in Fig. (6.2.1-e). A further decrease of T- results in oscillogram (c). In (d)

an arbitrary choice of "m" and "T-" was once more made. By decreasing "m" the

oscillogram in (f) resulted. Thus by varying any one of the variables m, b or T.,

a response of the form shown in Fig. (6.2.1) (e) may be obtained. This is not surpri¬

sing, since a first-order system has only one degree of freedom. If "m" represents

the maximum torque which can be applied to the system, then the response is the

optimum one.

In the following chapter a study of the effect of input modification on the res¬

ponse of second-order systems is presented.

100

0-t,

(a)

mn

i-(d)

(b) (e)

-tei(c)

<y++n-

(0

Fig. (6.2.1) Oscillograms of the Response of a First-Order System and its

Derivative to various forms of Inputs

(a) r(t), c(t)

(b) rm(t), c(t)

(c) rm(t), c(t)

(d) rm(t), c(t)

(e) r_(t), c(t)

(f) cw» gp-

- 101 -

(a)

•** —

(d)

(b)

i.

(e)

(c)

,m Tiff

(0

Fig. (6.2.2) Oscillograms showing the Effect of "m" and '"iy on the SystemResponse in the Phase Plane

(a) Arbitrary choice of "m" and "T " refer to Fig. (6.0.2)

(b) "Tj" decreased

(c) Further decrease of '"iy(d) Arbitrary choice of "m" and "Ty1(e) "m" decreased

(f) Further decrease of "m"

- 102 -

Chapter VII

Optimum and Quasi-Optimum Control. Second-Order Systems

7.0.0 Introduction

Systems whose behaviour may be described by second-order differential equa¬

tions occur very often in science and engineering, also the behaviour of many other

systems can be approximated to that of second-order systems. For these reasons,

on extensive amount of research has been directed in the past to study the charac¬

teristics of such systems.

In the field of automatic control, linear second-order systems play an impor¬

tant role. The transfer function of such systems is characterized by having two po¬

les. These poles may be either real or complex. In most cases, the system to be

controlled is fixed, that is, one cannot change any of its parameters; nevertheless,

it is usually required that the system output to a given input must satisfy a speci¬

fied performance criterion. This is, however, seldom possible. The procedure

usually followed, is to design a compensator which, when incorporated into the con¬

trol system, gives a system response that satisfies the performance criterion. Nor¬

mally a linear compensator is employed so that the final closed loop system remains

a linear one. A linear compensator, if well designed, is usually the best compensa¬

tor obtainable if the system is to be subjected to signals having a large range of

amplitudes or to signals of a large variety. However, if the variety of the input sig¬

nals is very restricted, for example, ramps only or steps only and especially if the

range of amplitudes is restricted, then it is usually true that performance of strictly

linear systems may be improved by applying nonlinear compensation techniques.

As has been previously mentioned in the introduction to chapter (3), a purely

linear system is subject to limitations. Linear operation rarely makes use of the

peak power capabilities of the system, also if use of the peak power is made, then

it is mostly for only a short portion of the operating period. This is always true for

small amplitude input signals. Small disturbances occuring in control systems are

corrected no faster than large disturbances. The transient response of linear sy¬

stems persists for infinite time, while that of nonlinear systems may be made of

finite duration theoretically as well as practically. A drawback of nonlinear compen¬

sation is that, while it might improve system response to one type of input signal,

- 103 -

it usually results in a response poorer than that of the original system when other

types of inputs are used.

In this chapter methods of compensating second-order open-loop control sy¬

stems are presented. Some of these methods are linear, but the majority is non¬

linear. In most of the work that follows only step inputs are considered. Such in¬

puts are frequently encountered in the control field, also because of their bandwidth,

they are very suitable as test signals. In the next section compensation is performed

using pulse modulation techniques.

7.1.0 Compensation using Pulse Modulation

7.1.1 Introduction

Pulse-width modulation techniques have been widely used in communication

and telemetry systems [16]. Recently these techniques have also been introduced

to the field of sampled data control systems. The first paper, to the author's know¬

ledge, was by Nease [24] of M. I. T. in which he considered an ideal pulse-width

modulator occuring in a single loop SDCS. He outlined linear compensation methods

based on approximation of the pulse-width modulated signal with impulses for small

signal stability and with the saturated signal for large signal stability. In 1958

Andeen [1] analysed a pulse-width modulated system where the pulse-width was a

linear function of the input. Kadota [18] applied the technique of "multipole expan¬

sion" used in electromagnetic theory with proper treatment of the Dirac delta func¬

tion to the study of magnetic amplifiers. Nelson [25] investigated pulse-width-con¬

trol as a method for on-off control of SDCS, pulse-width control as a means of compen¬

sating sampled relay controller and he also showed the analogy between pulse-width

control and saturating-amplitude control. His work is a great achievement, however,

it requires complicated mathematical description of the system. Billawala [3] showed

experimentally that pulse-width modulation had a damping effect and suggested its use

for obtaining deadbeat response and replacing conventional control circuit compensa¬

tors. Other papers on the subject of pulse-width modulation have also been publi¬

shed and some of them are given in the bibliography [2,8,10,11,19 ].

The difficulty in analysing or synthesizing control systems encorporating a

pulse-duration sampler, lies in the fact that in contrast to a pulse-amplitude sampler,

- 104 -

a pulse-duration sampler is a nonlinear element [11,24]. Two types of nonlineari-

ties are present:

a) The principle of superposition does not apply, that is, the sampled sum

of two inputs does not equal the sum of two sampled outputs corresponding

to these inputs.

b) The sampler output saturates for h = h,where h = T. T is the

iiiox* iiiajk*

sampling period and h is the variable pulse-duration, see Fig. (7.1.1).

As a consequence of these nonlinearities, the mathematical analysis of pulse-

width modulated sampled data control systems is not a simple one. For the exact

analysis of sampled-data control systems with finite pulse-width, the p-transform

method may be used [17]. It is essentially a step by step process of superposing

responses to individual input pulses and expressing them in infinite series so that

the final result may be written in a closed form.

If RD(s) is the p-transform of the input signal, G (s) the transfer function of

the system, then C (s) the Laplace transform of the response is given by

C(s) = Rp(s) G(s) (7.1.1)

Rp(s) is composed of functions of "s", of "e~ s" and of "e~ s". This implies that

the relation between C (s) and "h" is a complicated one, meaning that if the pulse-

u(t)'

1

0

7

^^

*

T T T* •

T

^J.

Fig. (7.1.1) Pulse-Width Modulation

u (t) the switching function

h1 , pulse-width

T sampling period, h = T

- 105 -

width "h" is varied one cannot easily determine the corresponding variation in

C(s).

To bypass the difficulties mentioned above, the author chose to work in the

time domain. By so doing and with the aid of the theory developed in chapter (H)

satisfactory results were obtained and most important, other methods of compensa¬

tion were discovered. The problem which was considered may be stated as follows:

7.1.2 Statement of the Problem

Given a linear system whose transfer function is known, it is required to

compensate the system in such a way that the response of the system to a specified

input satisfies some given performance criteria.

7.1.3 Solution

Let <j(s) be the transfer function of the given linear system. Let r(t) and c(t)

represent the input and the output of the system respectively. Denote the system

output that satisfies the performance criteria by c (t), that is the required system

output. Such an output can be obtained by inserting in the path of the input signal a

modifier that transforms the input signal r (t) into another signal r (t) having the

property that if introduced to G(s) gives c (t) as an output, see Fig. (7.1.2).

In this section, the modification will be performed by means of a computer

controlled sampler as shown in Fig. (7.1.3). The output of the generalized sampler

used is denoted by rms(0- The purpose of the computer is to operate the generalized

r(»M

R(s)

rm(t)

Hn(s)G(s) 4j.(t)

Cr(s)

Fig. (7.1.2) Linear System and Modifier

c (t) required output signal

M a modifier that transforms r(t) to r (t)

G (s) a linear system

- 106 -

"fc(t)

0-R(s)

rms(t)

FUs)G(s) _£(t)

C(s)

Fig. (7.1.3) Linear System and Computer Controlled Generalized Sampler

1 digital computer

2 a generalized sampler capable of

(a) Operating with a variable sampling period and/or variable

sampling duration

(b) Amplifying and/or attenuating the signals introduced to its

input

(c) Producing outputs having fixed amplitudes

3 a linear system having transfer function G (s)

r (t) the output of the generalized sampler and its Laplace transform

is Rms(s)f (t) the signal that controls the behaviour of the generalized sampler

sampler in such a way that its output r (t) has the same dynamic properties as

r (t), as governed by the Principle of Equivalent Areas developed in chapter (n).

H T denotes the sampling period, then r__(t) must satisfy the following re¬

lation

T

;nTn-1

r (t) dtmsv '

T

= ;nTn-1

rm(t) dt (7.1.2)

Equation (7.1.2) holds for the most general case where the sampling period as well

as the pulse duration may be varied. Clearly, this equation would still be valid un¬

der less general conditions, for example, where the sampling period is fixed and

the sampling duration is allowed to vary. A tabulation of different types of samplers

is presented in Fig. (7.1.4). The generalized sampler can amplify as well as attenu¬

ate signals; it can be imagined as having a variable amplifier in series with it.

- 107 -

The amplification in case of conventional samplers may be taken to be a constant

equal to one. The type (1) sampler, see Fig. (7.1.4), is the conventional sampler

used in finite pulse-width sampled data control systems [17]. The generalized

sampler when operated in mode (8), with the amplifier so adjusted that the samp¬

ler output has a constant amplitude and with the sampler pulse width linearly pro¬

portional to its input, is then identical to the pulse-width modulator referred to in

the literature [10,19].

Parameter Mode C V Mode C V

T

h

K

1

X

X

X

5

X

X

X

T

h

K

2

X

X

X

6

X

X

X

T

h

K

3

X

X

X 7 X

X

X

T

h

K

4 X

X

X

8

X

X

X

Fig. (7.1.4) A Tabulation of Types of Samplers

T Sampling period

h Pulse-width

K Amplification

C Constant or discrete

V Variable

x Indicates the existence of a state of operation

To illustrate how these samplers can be used to compensate control systems,

an example is given showing how a type (3) sampler may be used to compensate an

underdamped second-order system such that its response to a unit step input is a

deadbeat one.

- 108 -

Let G(s) be the transfer function of an underdamped second-order system and

let 0 (s) be given by

where a- and a are constants. The two complex poles of G(s) are s-0and occur

10 xt &

at Sj 2= - ccl j (J.

Let R(s) represent the Laplace transform of the unit step input,

R(s) = \ (7.1.4)

C (s), the Laplace transform of the response is given by

C(s) = G(s) . R(s) (7.1.5)

ora-s + a

C(s) =

s(s+oc+jp)(s+a-jp)<7-1-6)

Denoting the inverse transform of C(s) by c(t), therefore

.f a,s + a "I

c(t) = h'1 i1

2

° [ (7.1.7)L s (b+ «)2 + p

2 J

It can be shown that [7, pp • 90,91] c(t) may be written as

ao[(a -a-oG^ + ^p)2]! e"at

c(t)=-2. +—2-1 JL sin(pt + V) (7.1.8)

BZ PPo1 o

-1 al P -IP

where t > 0 and where ">u = tan - tan —'—T a -a, «. - oc

o 1

In terms of a specific example, suppose the system whose response is to be impro¬

ved is the series RLC circuit shown in Fig. (7.1. 5). Its transfer function G(s) is

given by

G(s) =5—- (7.1.9)

LCs + RCs + 1

109

TLTCS +TCS+1

where T.=^

and T„ = RC.

(7.1.10)

G(s)w.

~2 5"s + 2 £(*> s + u>

i o o

(7.1.11)

.2 1

L C

R Cwhere o)_ =

p=

—^and ^ =

r yr

The poles of G(s) occur at

\, i { i * Y<&*-A} (7.1.12)

Choosing some numerical values, let R = 0. 2 ohms, L = 1 henry and C = 4 farads,

therefore

G(s) = j—±4(s^ +0.2 s + 0.25)

^ = 0.2, wq = 0.5 and Sj 2= -0.1 t ] 0.49.

(7.1.13)

r(t) C± c(t)

Fig. (7.1. 5) An RLC Circuit used to simulate a Second-Order System

Substituting this data into equation (7.1. 8) the following is obtained:

-0.11c(t) 1 + 1.02 e sin (0.49 t - 1.77) (7.1.14)

c(t) is plotted in Fig. (7.1.6) curve "a".

- 110 -

It is required to compensate the underdamped second-order system such that its

unit step response is a deadbeat one. Denote the required deadbeat response by

c (t). Suppose that C (s) is given byr r

C(s) =i

2- (7.1.15)r

4s (s+0.5)"

Taking the inverse Laplace transform of equation (7.1.15) c (t) is obtained. This

was performed and the result is shown in equation (7.1.16).

cr(t) = 1 - e-0' 5 *- 0. 5 t e"°- 5t

(7.1.16)

c (t) is plotted in Fig. (7.1.6) curve "b".

To obtain c (t) as the output of the underdamped second order system, the

input to the second order system must be r (t). The Laplace transform r (t) is

Rm<s)m

and is given by:

Cr(s)G(s)

s2 + 0,

4 s(s

,2 s + 0.

+ 0.5)225

(7.1.17)

- Ill -

18 20 22 2*

Fig. (7.1.6) Step Response and required Step Response of a Second-Order System

c (t) step response of an underdamped second-order system

c (t) required step response of second-order system

Solving for r_(t), it can be shown that

r (t) = 1 - 0.8 t em

0.5t(7.1.18)

where t >0. rm(t) is plotted in Fig. (7.1.7).

A type (3) sampler, see Fig. (7.1.4), will be used to transform the signal r (t)

into r (t) which has the same dynamic properties as r (t). Li Fig. (7.1.8) theiiis m

block diagram of the system and controller is given. The digital computer is used to

compute the sampler pulse duration using the relation.

nT (n-l)T+k/ rm(t)dt = /

n

(n-l)Tm

(n-l)Tr(t) dt (7.1.19)

which expresses the Principle of Equivalent Areas for this case. The digital computer

then gives the orders to open and close the sampler according to the values of h cal¬

culated, h is the sampler pulse duration and is allowed to vary, h = T sees,

n is an integer greater or equal to one. T is the sampling period and is constant. T

must satisfy the condition

- 112 -

T £

*T^r

Til second.

(7.1.20)

(7.1.21)

V/hen r (t) is a constant, as is the case in the example considered, then equation

Fig. (7.1. 7) The required Modified Input r (t)

rpR(s)

'f

rms(t)

Rms(s)G(s) ^c(t)

*(s)

Fig. (7.1.8) Linear System and Computer Controlled Sampler

1

2

3

f

r(t)

msv'

digital computer

a type 3 sampler, refer to Fig. (7.1.4)

underdamped second-order system

signal controlling the behaviour of the sampler

the sampler output

- 113 -

nT

/ r (t) dt = (constant) . h (7.1.22)(n-l)T

m n

For r(t) = u -(t) and for r (t) given by equation (7 1.18), h may be expressed as

hn =TuAtn J (1 " 0.8te"0-56) dt (7.1.23)

n lu-lWl (n-l)T

where n "i- 1 and is an integer, lu j(t)| = 1 and T < 1 second. Equation (7.1.23)

when evaluated gives:

hn = t + 3.2 e"°*5t (0.5t + 1)nT

sees. (7.1.24)(n-l)T

n is an integer and is greater or equal to one, and Til second.

h has been evaluated for several values of T. The results are given in Fig.

(7.1.9). A sample of the calculations performed is presented:

For T = 0. 5 seconds and using equation (7.1.24)

hj = 0.5 + 3.2 e'°-25(0.25 + 1) - 3.2

= 0.5 + 3.2(0.78) 1.25 - 3.2

= 0.5 + 3.15-3.2

= 3.65 - 3.2

= 0.45 seconds

T = 0. 5 sees.

hj =0.45 hg =0.29 h17=0.45

h2 =0.27 h10=0.33 h18=0.46

h3 =0.22 hu=0.35 hi9=°-47

h4 =0.21 h12=0.37 h20=0,47

hg =0.21 h13=0.39 h21=0.48

hg =0.22 h14=0.41 h22=0-48

h? =0.25 h15=0.42 ^S^-48

h8 =0.27 fc^-44 h24=0-49

- 114 -

T = 1.0 sees. T = 2.0 sees.

hj =0.71 hj =1.16

h2 =0.45 h2 =0.95

h3 =0.44 h3 =1.34

h4 =0.51 h4 =1.7

hg =0.63 h5 =1.84

h6 =0.72 h6 =1.93

h? =0.80

hg =0.86

T = 3. 0 sees.

hg =0.90 hj =1.59

h10=0.94 h2 =1.95

hu=0.96 h3 =2.56

h12=0.97 h4 =2.86

Fig. (7.1.9) "h " for various values of "T"

hj = 1.16

h2 = 1 + 3.2 e~0,5(0.5 + 1) - 3.65

= 1 + 3.2(0.61) 1.5 - 3.65

= 3.92 - 3.65

= 0.27 seconds.

Once h have been determined, the response at any time "t" may be calculated by

superposing the responses to unit steps placed at t = (n-l)T and the responses to nega¬

tive steps placed at t = (n-l)T + h,where n is an integer greater or equal to one. As

an illustration for T = 0. 5 seconds and using equation (7.1.14), c(8) may be found as

shown in Fig. (7.1.10). In Fig. (7.1.11) c (t) is plotted for several values of the samp¬

ling period T, also plotted in the same figure is c (t) for comparison. From the latter

figure it is found that the smaller the sampling period T the better is the similarity bet¬

ween c(t) and c (t). The maximum deviation between c(t) and c (t) is:

T = 4. 0 sees.

hj =2.10

h2 =3.00

h3 =3.76

- 115 -

A B C D E

0.0 1.38 0.45 7.55 1.43

0.5 1.45 0.77 7.23 1.48

1.0 1.50 1.22 6.78 1.51

1.5 1.53 1.71 6.29 1.53

2.0 1.52 2.21 5.79 1.50

2.5 1.47 2.72 5.28 1.44

3.0 1.39 3.25 4.75 1.33

3.5 1.28 3.77 4.23 1.20

4.0 1.13 4.29 3.71 1.04

4.5 0.96 4.83 3.17 0.85

5.0 0.78 5.35 2.65 0.63

5.5 0.58 5.87 2.13 0.45

6.0 0.41 6.39 1.61 0.30

6.5 0.27 6.91 1.09 0.15

7.0 0.12 7.42 0.58 0.06

7.5 0.05 7.94 0.06 0.01

8.0 0.00 8.45 0.00 0.00

Sum 15.82 14.91

Fig. (7.1.10) The Evaluation of c(8)

The time at which the unit positive input steps occur

The magnitude of the response at t =8.0 sees, for the steps occuring

at time A

The time the unit negative steps occur. This time is equal to (n-l)T+h

where (n-l)T = A

(8-C) sees.

Magnitude of the response at time t = 8.0 sees, for the negative steps

occuring at time D.

c(8) = 15.82 - 14.91 = 0.91

- 116 -

2% for T = 0. 5 seconds

5% for T = 1. 0 seconds

7% for T = 2.0 seconds

16% for T = 3.0 seconds, and

22% for T = 4.0 seconds.

fl was pointed out in equation (7.1.21) that to obtain a faithful reproduction of

the required output c (t), T must satisfy the condition Til second. This checks

with the results found above, since 5 % is usually an acceptable tolerance.

With no sampling the output c (t) required more than 25 seconds to arrive to

within 5 % of its final value, see Fig. (7.1.6), while when a sampler whose pulse-

c (t)*

Fig. (7.1.11) c(t) for various Values of the Sampling Period T

1 T = 0.5 sees. 4 T = 3.0 sees.

2 T = 1.0 sees. 5 T = 4.0 sees.

3 T = 2.0 sees. 6 T = cr(t)

width is controlled is encorporated in the path of the input step, it took 10 seconds

for the response to arrive to within 5 % of its final value using a sampling period of

T = 0.5 seconds.

The example which has just been solved is an illustration of how compensation

may be achieved using a sampler which operates according to the Principle of Equi¬

valent Areas. Other techniques for system compensation using switching also exist.

In the following section some of these techniques will be studied and further developed.

- 117 -

7. 2.0 The Impulse Modifier

The method of compensation presented in the previous section may be summa¬

rized as follows:

a) Determining the input to the given linear system which produces the requi¬

red output.

b) Modifying the input to the linear system using a sampler so that the output

of the sampler - which becomes the new input to the system - has approxi¬

mately the same dynamic characteristics as the required input.

The methods of compensation which will now be presented are best understood

by means of examples. From the problem considered in the previous section, it may

be concluded that to obtain a system output c(t) having a short rise-time— where the

rise-time of a signal is defined as the time required for the signal to go from 10 %

to 90 % of its final value - the modifying signal must have a large magnitude and a

short duration. By modifying signal is meant the signal which when subtracted from

r (t) gives r (t). If the rise-time is required to be the same as that of the step res¬

ponse of the underdamped second-order system, then the modifying signal must ap¬

proach its limit which is an impulse. To check the truth of this suggestion, let

c (t) shown in Fig. (7.2.1) be the required response. This response may be obtained

by subtracting from c (t) - the step response of the underdamped second-order system

the dotted signal cJ.t)- If c ,(t) can be obtained as the system response to an impulse,

then the suggestion made is true. In terms of the example previously considered, the

impulse response c.(t) is given by

ci« " L_1 [(s+oc+ipMUoc+jp) <"-D

= L_1 [i+^f + B^kp] <7-2'2>

= Ajet-^P)* + ^-""IP* (7.2.3)

*i - sT^nr J = s#- - $re_j x <7-2-4>

s=-et+jfi

- 118 -

A« is the conjugate of A. and is therefore given by

Substituting equation (7.2.4) and (7.2. 5) into equation (7.2.3) gives

(7.2.5)

Cdd), fCr(t)

1.6

\\Cd(t)\\ Cr(t)T

s/

-/-\

\/

\v_

•

-\z. £ • 1-

0 2 A 6 8 10 12 14 16 18 20 22

Fig. (7.2.1) Decomposition of c(t) into c.(t) and c (t)

c.(t) signal given by equation (7.2.10)

c (t) required step response of underdamped second-order system

ao -«t TX

or

Cj(t) = £ e-~lcos(pt--^-)

ci(t) =

T •""'"'"P*

(7.2.6)

(7.2.7)

For aQ= 0.25, (i = 0.49 and ot = 0.1 as in the example considered in the previous

section, equation (7.2.7) becomes

c.(t) = 0. 51 e"0, U sin 0.49 t (7.2.8)

- 119 -

On plotting equation (7.2.8) and comparing it with c.(t) given in Fig. (7.2.1) the only

difference found was a constant multiplier. This constant multiplier was later found

to depend upon the relative damping | . That is, it was found that

cd(t) = X(?) - c.(t) (7.2.9)

where

cd(t) = cft+tj) - 1 (7.2.10)

"t" is the time required for the system unit step response to reach a magnitude of

one for the first time, see Fig. (7.2.1).

From equation (7.2.9) and using the numerical values given in Appendix A, it

was found for example that

Wn „. 0. 5266000X(0-2)

=

6.3780700

= 1.391 (7.2.11)

To confirm the validity of equation (7.2.9), three points will be checked.

For \ = 0.2, t« = 3.617 sees, as may be shown using equation (7.1.14) or from

the tables in Appendix A. Therefore, using equation (7.2.10)

cd(2) = c(2-t5.617) - 1

= c(5.617) - 1

= 1.483 - 1

= 0-483 (7.2.12)

From Appendix A

c.(2) = 0.3469400 (7.2.13)

Substituting equation (7.2.11) and (7.2.13) into equation (7.2.9) gives

cd(2) = 1.391 (0.3469400)

= 0-483 (7.2.14)

- 120 -

Following the same procedure used above

cd(S) = c(3+3.617) - 1

cd(3) = c(6.617) - 1

= 1.524 - 1

= 0.524

c.(3) = 0.37612

Ich gives

cd(3) = 1.391 (0.37612)

= 0.524

cd(l) = c(1+3.617) - 1

= c(4.617) - 1

= 1.303 - 1

= 0.303

(7.2.15)

(7.2.16)

(7.2.17)

(7.2.18)

Cj(l) = 0.2172700 (7.2.19)

Therefore,

cd(l) = 1.391(0.2172700)

= 0.302 (7.2.20)

From the above results it may be concluded that equation (7.2.9) holds.

Using equation (7.2.9) and with the aid of the tables given in Appendix A the

following values of X( t ) may be obtained:

vm ?^ -

0- 5266000_ t „q1X(0-2) -

0.37S07OO~ 1-391

v/n 9^ -

°-444300_

X(0-25) 0.355740~ L249

X(0-30) -

0.335690- 1,1U

- 121 -

X(0.35) = 0.975 X(0.40) = 0.842

X(0.45) = 0.715 X(0.50) = 0.597

X(0.55) = 0.485 X(0.60) = 0.380

X(f ) versus $ is plotted in Fig. (7.2.2).

a may now be concluded that to obtain a response c (t) as shown in Fig. (7.2.1),

the input should be a unit step starting at t = 0 with a negative impulse of magnitude

X(l[) occuring at t = t* superposed on it. What happens in the phase plane is roughly

sketched in Fig. (7.2.3).

In (a), the unit step and the negative impulse of magnitude X(1J ) responses are

shown. Li (b), a sketch of c (t) is given. A formula expressing X(^) in terms of

1J and co may be derived as follows:

The unit impulse response of the underdamped second-order system is

c.(t) = -£ e'^sinpt (7.2.21)l p

2 2 2where a = P0

= (<*• + P )• Equating the derivatives of this equation to zero, the

time t„

at which c.(t) has its first maximum is found to bemax. i

*!.»«• = ^-tan_1-^- (7.2.22)

For the unit step response

, 2 2.1/2

'peak =(0C

V}

* <*-2-23>

C<W -1 = e" -T11 =

em(t)...(say) <7'2-24>

Therefore,

X(ee, p) = °^Peak) r1

(7.2.26)l^imax.'

.J5.H

= g, P} (7.2.27)

- 122 -

1.2

1.0

08

06

0A-

0.2 +

0

Q1 02 03 04 0.5 06

Fig. (7.2.2) X( If) versus \

^ the second-order system relative damping

X ( ^ ) a function of ^ defined by equation (7.2.9)

c(t) cd(t) cr(t)

cW.-q/t) 0 cr(t)

Fig. (7.2.3) The Behaviour of c(t), cd(t) and cr(t) in the Phase Plane

c(t)

cd(t)cr(t)

underdamped second-order system step response

signal given by equation (7.3.10). It is shown dotted in (a)

required step response

c.(tr max.

2„2

PC + ft-eel-tan-1^.

I* * sin (J — tan (7.2.28)

Therefore,

- 123 -

B-i fit-tan-1!-]

Y/„, n. _P e fi oc J

(oe + [O sin tanX -§-

In terms of £ and to

(7.2.29)

o

-1

VT>2

t.„ = -1 tan-1 Vl '

* (7.2.30)

-„- t

upeakt .

=

*(7.2.31)

1(7.2.33)

emW = e-( ,j _„"> (7.2.32)

and

therefore,

X(^,UJo)

Jl/ 2" 2W (7T " tan_1 "^TVl - ^

2e (1-1?T/2 *

(7.2.34)

-1 \ 1 - Xoo sin tan — —->-

° X

In Appendix B it is proved that X( ot, ft) given by equation (7.2.29) - or equiva¬

lents by equation (7.2.34) - when multiplied by c.(t) results in c .(t).

To obtain the theoretically fastest step response to an underdamped second-or¬

der system, the modifying signal must be composed of an impulse and a doublet, this

can be shown as follows:a

Suppose the required output is a step of magnitude r—k- and that the system"°

transfer function is that given by equation (7.1.6) and since

C(s) = R(s) • G(s) (7.2.35)

therefore,

- 124 -

—JSL = R(s) JS T (7.2.36)

(3 s (s+oc) + fi

2 2 2But a = (3 = oc + p , consequently

a

R(S) =

—-sr ( — )sZ + 2 PCs + oc + P'

I o2

(3,

(s + 2 «+ -^)-ijS

Po

1s + ^ +

i (7.2.38)

Po Po

Therefore the required input is

r(t) = -L u,(t) + ^ a(t) + u At) (7.2.39)

pI p!

which is seen to be composed of a doublet of magnitude —ij- ,an impulse of magnitude

2<x, P°zy

and a unit step all situated at the origin of time.

P°

7.3. 0 The Pulse Modifier

If the Principle of Equivalent Areas developed in chapter H is a valid principle,

then it should be possible to replace the impulse used as the modifying signal in the

previous section with a pulse of equivalent area and obtain a similar response. This

was done and the results obtained were very satisfactory. Some of these results will

be, presented followed by a graphical procedure which facilitates the determination

of the appropriate modifying pulse. Next, the condition that the pulse duration must

be small compared to the dominant time constant of the system will be relaxed and

negative pulses having longer durations will be used to modify the input step so that

the response is a deadbeat one. Further, a relation will be obtained between the areas

of these pulses, their location and magnitude.

According to equation (7.2.9), the magnitude of the modifying impulse that gives

a response c (t) as shown in Fig. (7. 2.1) is X units and for | = 0.2 it was found that

- 125 -

X(0.20) = 1.391. Therefore, the area of the narrow pulse which is to replace the

impulse must also be 1.391 units of area for it to be dynamically equivalent to the

impulse. If the amplitude of the short duration pulse is only allowed to have a value

equal to that of the amplitude of the input function and a value zero, which implies

in the case considered an amplitude of unity and zero, then the duration of the short

pulse denoted by t,will be (t -1.391) seconds. Such a modification can be perfor¬

med by means of a simple switch which opens at the appropriate time and closes t

seconds later. The modified input may then be written as

rm(t) = u_x(t) - u.^t-tj + 0. 5tp) + u^t-tj - 0. 5tp) (7.3.1)

= u_j(t) - Ul(t-3.6174 + i^i) + u 1(t-3.6174-i-|?l)

= u_j(t) - u_j(t-2.9219) + u_1(t-4.3129) (7.3.2)

where tj is the time the unit step response of the second-order system has a magni¬

tude of one for the first time. To facilitate the numerical calculations, an r (t) given

as follows will be used:

rm(t) = u_1(t)-u_1(t-2.9)+u_1(t-4.3) (7.3.3)

The error in t which results by using r (t) of equation (7.3.2) rather than that

of equation (7.3.3) is o. 9 %. The corresponding output c (t) has been calculated

using superposition, for example:

crm(5.5) = c(5.5) - c (5. 5-2.9) + c(5.5 - 4.3)

= c(5.5) - c(2.6) + c(1.2)

= 1.469300 - 0.623800 + 0.1615700

= 1.00707

crm(t) is plotted in Fig. (7.3.1). Its maximum overshoot is 0.8 %, which is a satis¬

factory result considering the 0.9 % approximation made in the choice of r (t). Thus

it may be stated that here also the Principle of Equivalent Areas is valid and that the

- 126 -

modifying impulse may be replaced by a narrow pulse having the same area.

A graphical procedure is now presented which is very simple and which pro¬

vides an alternate way of determining the location of the switching instants t,t so

that the response is deadbeat as shown in Fig. (7.3.1).

The procedure may be stated as follows:-

a) Determine t«, t„, t, from the plot of the unit step response of the under-

damped second-order system, see Fig. (7.3.2).

b) Measure |e (t)|.c) Determine to - t„.

d) Superimpose two plots of c (t) - the unit step response of the system - on

each other as shown in the figure.

e) Move the step response curves with respect to each other till lK(x) I

+m<4 where t

f) Measure t and t .

x z

g) Thent

h - V

*z and *b = *3 K + t.

Z X

CrmW*

6 8 10 12 14 16 18 20 22

Fig. (7.3.1) The Step Response c (t) obtained using r (t) of Equation (7.3.3)

- 127 -

c(t)1,6

Fig. (7.3. 2) A Graphical Procedure for determining the Switching Instants so that

the Step Response is Deadbeat

Using the above procedure and starting with two curves of the unit step response

of the underdamped second-order system having E = 0.2 and cj =0.5 rad. /sec., it

took the author less than three minutes to estimate the modifying input as being

rm(t) = u_x(t) - u_j(t - 2.8) + u_j(t - 4.2) (7.3.4)

Such a modifying input gave a response with a maximum overshoot of 3% which is

within normal engineering tolerances and is a good result considering the errors in

plotting and in reading the curves.

Another graphical method for determining t and t consists of plotting on milli¬

meter paper a small portion of the unit step response of the underdamped second-or¬

der system starting from the instant its magnitude is one for the first time, i. e. star¬

ting from tj shown in Fig. (7.2.1). On a second sheet of millimeter paper a small por¬

tion of the unit step response starting from zero time is then plotted. The two sheets

of millimeter paper are placed on top of each other so that the curves are as seen in

Fig. (7.3.3). One of the latter sheets of paper is then moved with respect to the other

until the curves are tangential, see curves 2 and 3 ind Fig. (7.3.3). (t-t ) is given

by:-

- 128 -

C(t)

t !la

Fig. (7.3.3) A Method of Determining the switching Instants to give a Deadbeat

Response

1,3 portion of step response of underdamped second-order system star¬

ting from the time c (t) =0

2 portion of step response starting from the time c(t) = r(t), that is,

from time t = tp see Fig. (7. 2.1)

t a time interval as seen in figure

Vla = 2tq <7-3-6>

where t is as shown in the figure. As mentioned previously, (t- - t ) = (t - t,),

therefore,

*!-*» = *b-*l * *q (7-3-6)

This graphical procedure Is also easy to apply, however, it has a disadvantage

which is, that it is usually not easy to determine exactly when the two curves are tan¬

gential, consequently appreciable error in the t determined sometimes exists.

The type of pulse switching which has been described has been studied in the

laboratory. The VSSG was used to provide the modified input and the underdamped

second-order system was simulated by means of an RLC network. Some of the re¬

sults obtained are shown in Fig. (7.3.4). In (a), the step response of the underdamped

second-order system is shown. In (b), the modified input r__(t) and the correspon¬

ding output are given. Oscillogram (c) shows the modified input and the step response

of the underdamped second-order system. In oscillogram (d), the responses shown

in (a) and (b) are given in the phase plane. In Fig. (7.3.5) the effect of an incorrect

choice of the modifying pulse is indicated in the phase plane. In (a), the amplitude

of the modifying pulse was too small and in (b) it was too large. In (c), the modifying

- 129 -

\ Y

/(a)

|

1Ml

f 4

/ f

1111 J V ijj n^i

1

(b)

Fig. (7.3.4) Effect of narrow Pulse Modification on the System Response

(a) r(t)andc(t)

(b) rm(t)andc(t)(c) rmW and c W °' case (a)

(d) Behaviour of the response shown in (b) in the phase plane

130 -

&^

(a)

(<0 (f)

Fig. (7.3.5) Effect in the Phase Plane of Incorrect Choices of the Modifying Pulse

7.4.0 Posicast Control

The modifying pulse used in the previous section does not necessarily have to

have a duration which is small compared to the period of oscillation of the system

in order that a deadbeat response be obtained. To show the truth of the latter state¬

ment and in general to help understand this method of control, consider the oscillo¬

grams shown in Fig. (7.4.1). In (a), the input is composed of a positive step and a

delayed negative step superimposed on it. As was shown in chapter VI, such an in¬

put has a pole at the origin of the s-plane and an infinite number of zeros parallel to

the j to-axis, refer back to chapter VI. If the instant the negative step is applied is

denoted by t,

its amplitude by K, and that of the original step input by K-

ao

J&'

- 131 -

then the response shown in Fig. (7.4.1), (a), may be written as

c(t) = k\ + K—^-e^sinfpt +V) --% Ki

Po PPo Po

^2_K]l e" "^"V sin [ fi (t-ta) + Tj)] (7.4.1)

PPo

The latter response - as may be seen from oscillogram (a) - is oscillatory. This is

because the maximum and the zero overshoots of the response - which correspond

to instants of maximum and zero potential energy of the system respectively - alter¬

nate and do not coincide with the maximum and zero rates of change of the overshoot -

that is, with the instants of maximum and zero kinetic energy of the system - till

after infinite time has elapsed.

H at any instant both the kinetic energy of the error and its potential energy

could be made equal to zero, then the system will remain at rest. A simple way of

achieving this is to apply an input to the underdamped second-order system of the

type shown in Fig. (7.4.1), (a), and at the instant c (t) =0 which will be denoted by

t., let the input to the system jump to a magnitude equal to c (t). The result of follo¬

wing the latter procedure is shown in oscillogram (b). In oscillograms (c) and (d) the

effects of applying the positive step too late and too early are given respectively, t

could be found by equation c (t) to zero and solving for t all other constants being known.

The magnitude of the final step may be calculated once c (t.) has been found. If the

magnitude of the final step is known beforehand, then to determine t two simultan¬

eous equations must be solved, where either t or Kt is known, noting thata j.

K - Kj + K2 = Kf (7.4.2)

where K, K. and K„ are the magnitudes of the first, second and third steps respecti¬

vely. K. is the magnitude of the required steady state output.

The calculations which must be performed usually require a great amount of

labour so that a graphical procedure similar to that described in the previous section

is probably more convenient not mentioning analog computer simulation and the use

of a VSSG as the one described in chapter m.

- 132 -

—

N ::

\ /r U

(a) (c)

++H +ttt* h+H

(b) (d)

Fig. (7.4.1) Analysis of narrow Pulse Modification

Oscillograms showing the inputs to an underdamped second-order system and thecorresponding outputs

133

r^ [—1

(—

m(

/ /

(a) (e)

ii,Cf 7

(b) (f)

7

(c) (g)

: t 1,-- "•K^^^— ^-it

*"ii+~

J(d) (h)

Fig. (7.4.2) Posicast Control using modifying Pulses having different Widths

(a).. (g) the inputs to an underdamped second-order system and the corres¬

ponding outputs

(h) system output showing the improvement of system response when

posicast control is used

Li Fig. (7.4.2), (a), the step response of an underdamped second-order system

is given. Oscillograms (b) ... (g) show how by selecting the appropriate modifying

pulse a deadbeat response may be obtained. The duration of the modifying pulses ran¬

ged from ones which are large compared to the period of oscillation of the underdam¬

ped system such as in oscillograms (b) and (c), to others which are small compared

to it such as in oscillograms (f) and (g). In (h), deadbeat and oscillatory responses

are given in one waveform.

- 134 -

If the modifying pulse is chosen to be equal to half the period of oscillation of

the system the response is then as shown in Fig. (7.4.3). In (a), "1" and "2" repre¬

sent the step responses of the system where "2" is delayed with respect to "1" by t,

see equation (7.2.31). When c,(t) reaches its peak at t,the input step is suddenly

raised to a magnitude equal to c«(t ). The resultant response is then deadbeat and is

shown in oscillogram (a) as "3". In (Jo), the modified input and the deadbeat response

are given. Oscillograms (c) and (d) show the behaviour of the oscillatory and the dead-

beat responses in the time domain and in the phase plane respectively.

An input signal as that given in Fig. (7.4.3), (b), has a z-plane zero, see [31].

This can be shown using the same method employed in chapter VI. Denoting the mag¬

nitude of the first step by m and that of the second step by d, the Laplace transform

of the modified input Rm(s) is given by

Vs) = i [m + de-BTl] (7.4.3)

where T, is the time the second step occurs after the application of the first step.

The zeros of equation (7.4.3) occur when equation (7. 4.4) is satisfied

de~sTl = -m (7.4.4)

Writing s in terms of a real and an imaginary component, that is, s = cc + j p, equa¬

tion (7.4.4) may be written as

e^l (cos fiTj + jsin (JT^ = - -^ (7.4.5)

By equating together the real and imaginary parts of equation (7.4. 5) « and (S are

found to be given by

(7.4.6)cc =

1i

d

T^to

m

p =+ (2n+l) Tl

n = 0,1,2 ... (7.4.7)

By a proper choice of the variables — and T, a pair of complex zeros of the modi¬

fied input may be made to fall on the poles of the underdamped second-order system

- 135 -

2

-"^ k 3

++++ +H+ -H++ +H+ +H+ **++ ttff1

TH+ ++++ +H ++++ +++f 4+H- -H4f +H+ +H+ ++« H+f ++++

/(a) (b)

c(t)

++ c(t)

Fig. (7.4.3) Half-Period Posicast Control

(a) 1: step response of second-order system2: step response of second-order system delayed with respect to the latter one

3: resultant of "1" and "2". Note: the zero-level in "2" was displaced for

convenience, however it should be as in "1"

(b) r (t) and the corresponding c (t)

(c) the step response with and without input modification

(d) the behaviour in the phase plane of the responses given in (c). For the

underdamped response the sign has been reversed.

- 136 -

thus cancelling them and resulting in a deadbeat response.

In Fig. (7.4.4) some of the effects of an incorrect choice of — and T1 for the

n = 0 case are shown. In (a) — was too large but T- was correctly chosen. In this

case the pair of complex zeros have the correct value of p but their real part oc is si¬

tuated in the complex frequency plane to the left of the poles of the underdamped se¬

cond-order system. Li (b) — was too small, but T- was correctly chosen. In (c)

— was correctly chosen, but T- was too small. The complex zeros generated lie

HH ++++ +ttt ++« ****. JIM

"1/

(a)

-?r*++ -" _'

7 |(b)

7\

(c)

/" /"4+tf

/

c(t)

h

(e)

h

(f)

_1_../ \

^

r±3[) VlX-J

(g)

s?

m„4~2 tin

^X

Tv A

c(t)

(d) (h)

Fig. (7.4.4) Effects of incorrect Choice of the modifying Pulse

(a).. (d) r (t) and the corresponding c (t)

(e).. (h) the behaviour in the phase plane of the latter responses respec¬

tively.

to the left of the poles of the system and have a (J larger in magnitude than that of

the underdamped second-order system. In (d) T- was too large, but — was correct¬

ly chosen. The behaviour of the responses shown in (a) ... (d) in the phase plane are

given in oscillograms (e) ... (h) respectively.

- 137 -

The response shown in Fig. (7.4.3), (b) was obtained as a result of the cancel¬

lation of the pair of complex poles of the underdamped system by the n = 0 zeros of

R (s). An R (s) is also conceivable whose n = 1 or n = 2 pair of complex zerosm m

cancel the pair of complex poles of the second-order system. Such modified inputs

and the corresponding responses are shown in Fig. (7.4.5), (a) and (b). In oscillo¬

gram (a), the complex poles of the system are cancelled by the n = 1 pair of complex

zeros and in (b) they are cancelled by the pair of complex zeros whose n = 2.

A response similar to that given in oscillogram (a) may be also obtained by

injecting a negative pulse at the appropriate instant, see oscillogram (c). The latter

type of input modification is the same as that described in the previous section. It

should be noted that if the final steady state level of the output is the same for the

responses given in oscillograms (a) and (c), then the oscillation of the response in

oscillogram (a) will have a smaller overshoot than that given in (c) but will require

a longer time to reach the steady state value than the time required by (c). The beha¬

viour of the response shown in oscillogram (c) in the phase plane is shown in oscillo¬

gram (d) together with the step response of the system for comparison. It might be

worth noting that the injection of negative pulses can cancel initially positive oscilla¬

tions and that the injection of positive pulses can result in the cancellation of initial¬

ly negative oscillations.

A response having a single overshoot may be also achieved by letting the comp¬

lex poles of the system be cancelled by the n = 1 zeros generated by a type of input

as that previously used in chapter VI. Some results are shown in Fig. (7.4.6). Oscil¬

logram (a) shows the system step response. Oscillograms (b) and (c) show how after

one overshoot a period of no oscillations may be achieved and oscillogram (d) was gi¬

ven to indicate that in general, step input modification can result in a variety of inter¬

esting - and may be in the future useful - waveforms.

In oscillograms (a) and (b) of Fig. (7.4.7), the input modification was achieved

by the injection of two negative pulses at the appropriate instants. Oscillogram (c)

shows a square wave whose positive step has been sampled at a high frequency, for a

certain interval of time. The latter waveform was generated using the variable-fre¬

quency on-off signal generator described in chapter in. The corresponding response

is given in (d). In (e), the response and rate of change of the response are shown for

an input modification of the type given in Fig. (7.4.3), (b). The behaviour of the res¬

ponse given in (e) in the phase plane is shown in oscillogram (f). In (g), the rate of

- 138

; :

; ;

1

r1 \y

;:

(a)

':':

+H+ 1++++T

++++ H+Mm ++*h+++H ++H +++

/

1 V

III 1

/ ::

^

(c)

(b)

c(t)

Fig. (7.4. 5) Cancellation of the complex Poles using higher-order Zeros

(a) r (t) and the corresponding c (t)

(b) r (t) and the corresponding c (t)

(c) r (t) and the corresponding c (t)

(d) the behaviour in the phase plane of the response given in (c)

139

/> r\/ i

\u"/ "7j

(a) (c)

++++

^+++H H++ +>tt±M^iH

rs•

+ IH-+W+-

+H+

(b) (d)

Fig. (7.4.6) Possible Responses

(a) step response

(b).. (d) modified inputs and the corresponding outputs

140 -

u.ft+t «++ t+++

H

s:

(a)(b)

(c)

M

tf V

(e)

(d)

c(t)

(0

/—"^^<-\ \

m^*£rc(t)

1 i

rr —: \n /- >vV '/> «J! j*4

(f0 (ii)

Fig. (7.4.7) Some Input Modifications which result in Deadbeat Responses

(a).. (c) r (t) and the corresponding responses respectively

(d) the response corresponding to r (t) given in (c)

(e)

(f)

c (t) and c (t) when an input as in Fig. (7.4.3), (b), is used

the latter response in the phase plane

(g) rm^ an<* *ne corresPonding c (t)

(h) a step input and its response

a modified input and the corresponding response

- 141 -

change of the second-order system response is given when the input is modified by

the injection of a short negative pulse as shown also in the same oscillogram, (h) is

given to show that responses which reach their steady state value in less than one

quarter of the period of oscillation of the system are possible to obtain. A step in¬

put, a modified input, the step response and the response corresponding to modified

input are all given in the same oscillogram.

In the following chapter a study is presented of the posicast method of control

for third, fourth and fifth-order systems.

- 142 -

Chapter III

Posicast Control Applied to Higher-Order Systems

8.0.0 Introduction

In the previous chapter, the posicast method of control was applied to second-

order systems. In this chapter, the posicast method of control will be applied to

third, fourth and fifth-order systems.

Smith in his book [31] suggested a procedure for designing the posicast com¬

pensator for higher-order systems. Zaborsky [37] investigated the effect of inputs

such as the one given in Fig. (7.4.3), (b), on lightly damped second-to fifth-order

systems. Lendaris and Smith [20 ] showed the effect of pole cancellations of a third-

order system by a zeros generator. So and Thaler [32] pointed out that if the stored

energy in the pertinent components of higher-order systems could be dissipated at

the switching instant, the response of the system will be improved. In the following

section the posicast method of control is applied to third-order systems.

8.1.0 Third-Order Systems

A third-order system is a system which has three poles. These three poles

could be either all real or one of them could be real and the other two complex. The

latter case is more interesting and so will be considered here, also the results for

the former case are readily deducable from those of the latter one.

In Figs. (8.1.1), (8.1.2) and (8.1.3) oscillograms (a) to (d) show the inputs to

the third-order system considered and the corresponding outputs respectively. In os¬

cillograms (e) to (h), the outputs are repeated and included are the outputs of the first

stage of the third-order system. The latter outputs shall be referred to as the c (t)

outputs. In oscillograms (i) to (1) the behaviour of c(t) in the phase plane is given.

As was shown in chapters VI and VH, by modifying a step signal in one of seve¬

ral ways the resultant modified signal will have z-plane zeros. In Fig. (8.1.4) the

poles of the system in the complex frequency s-plane are sketched, also shown are

the pole zero locations within the pass band of the system of the modified inputs con¬

sidered in this section.

- 143 -

In Fig. (8.1.1), (a), the step response of the third-order system is shown. Such

a system has a pole zero location as indicated in Fig. (8.1.4). The exact values of the

real and the imaginary parts of the poles are not important in this study, but rather

the relative positions of the poles and the zeros. The input step contributes a pole at

the origin of the complex frequency s-plane. Modifying the input step in the way shown

in (l.b), that is in Fig. (8.1.1) oscillogram (b), is equivalent to adding to the pole zero

pattern of the input step a z-plane zero. This z-plane zero contributes in the passband

of the system the pair of complex zeros shown in Fig. (8.1.4). This pair of complex

zeros coincides with the pair of complex poles of the system and so cancels them.

The behaviour of the system is then governed by the remaining real poles and

consequently the response is exponential as may be seen from oscillograms (l.b),

(1. f) and (1. j). To determine the location of the zeros, equations (7.4.6) and (7.4.7)

may be used.

The modified input given in (1. c) has its zeros to the left and above the complex

poles as shown in Fig. (8.1.4). These passband zeros are farther away from the ori¬

gin than the poles of the system and so are not effective. Consequently, the response of

the system to the modified input given in (l.c) is nearly the same as its step response.

In (1. d), the modified input has a large T-, see equations (7.4.6) and (7.4.7),

with the result that two pairs of complex zeros of the modified input lie in the pass-

band of the system. The estimated pole zero location is given in Fig. (8.1.4).

In (2. a) Tj is the same as that used in (l.b), which means that (5 of the pair

of complex zeros in the passband of the system is the same as that of the complex

poles, ofc, however, is much larger than that of the complexpoles of the system, since the

ration of — is larger than in case (l.b). The pair of complex zeros is also shown in

Fig. (8.1.4). The effect of reducing the ratio of — may be seen from (2.b). Maintai-

dm

ning a small ratio of — but introducing a negative step instead of a positive one as

shown in (2. c), the response is seen to be more oscillatory than that of the pure step

response. The zeros generated by the input modifier are indicated by "7" in Fig. (8.1.4).

Increasing the ratio of —,the response in (2. d) is obtained. The latter response is

highly oscillatory because the real pole of the system is nearly cancelled by the real

zero generated by the input modifier while at the same time the pair of complex zeros

have a negligible effect on the system response compared to that of the complex poles

since the latter are much closer to the origin than the complex zeros, see Fig. (8.1.4),"8".

- 144 -

\ \

(a)

/r-

/

/

(b)

(c)

y^—i V :!

JtC i

ft'X'^—Amm.m.J,^,^^

(e)

-*'

/;-

i

(0

(d)

c(t)

f

>J

(I)

i--

*,

"

K

:

a)

^j* 1

c ^ m,

*. ^

(k)

tC ^, V \'"( : 7

m,^*- ->

•-c(t)

(h) (1)

Fig. (8.1.1) Oscillograms illustrating Posicast Control for Third-Order Systems

a... d Inputs and the corresponding outputs

e... h outputs of the second stage and of the third-order system

i... 1 the behaviour of the outputs in the phase plane. The curve in the

lower half of the phase plane represents the response of the third-

order system to a negative step

145 -

«t=»

1

m>

/

(a)

{

(b)

f]f\

1

(c)

A

/>T\

/(d)

+4»4+tf

/

(e)

i

(0

^ £1

(g)

A

A i/n

vlA rv1

./'

'

(h)

<«~

c(t)

t

t**t

(0

(i)

(k)

- - c (t)

,'r*>

Um Prt ^ ( 1».

V }Jv*- >

(1)

Fig. (8.1.2) Oscillograms illustrating Posicast Control for Third-Order Systems

a... d inputs and the corresponding outputs

e... h outputs of the second-stage and of the third-order system

i... 1 the behaviour of the outputs im the phase plan. The curve in the

lower half of the phase plane represents the response of the third-

order system to a negative step

- 146 -

tjfltm

c(t)

(a)

y\*

(b)

M.

(c)

c(t)

<\

,/, ^~ anr rm ,J,y

i—'"\

1 / if

(d) (h) (1)

Fig. (8.1.3) Oscillograms illustrating Posicast Control for Third-Order Systems

a... d inputs and the corresponding outputs

e... h outputs of the second stage and of the third-order system

i... 1 the behaviour of the outputs in the phase plane. The curve in the

lower half of the phase plane represents the response of the third-

order system to a negative step

- 147 -

o14o10

08 09

03

07

04

011

05

14 10-o—o—

05

o 201112

@„o16

03

08 09

04

04

07

o10

AJU)

1?©13 o16 06

015

015

o15

o15

o15

.15

06

o15

015

015

°15

o14

Fig. (8.1.4) The Location of the Poles and Zeros of the Third-Order System and

of the Inputs to it

Poles at

0 of step infrat

2,9 of third-order system

Zeros at :

2

3

of modified input11 u 11

of Fig.11

(8.11

4 ti ti it tt II

5

6

11

11

n

it

tt

ti

"

Fig.II

(8.II

7 ti ti 11 II 11

8 11 11 ti It II

9

10

11

n

ti

ti

tt

it

" Fig.II

(8.It

11 t! 11 ti II tl

12 it 11 11 II II

13

14

11

ti

it

ti

n

11

" Fig.II

(8.II

15 11 n ti 11 11

16 it ti 11 "Fig. (8.

1.1),

1.2),

1.3),

1.5),

1.6),

(b)(c)(d)(a)(b)(c)(d)(a)(b)(c)(d)(a)(b)(f)(a)

- 148 -

Reducing the ratio of — so that the n = 0 zero generated by the input modifier

coincides with the real pole of the system the response is then as seen in Fig. (8.1.3),

(a). The third-order system behaves as an underdamped second-order system as may

be seen from its response to the modified input given in (3. i).

Decreasing T-, oscillogram (3.b), results in moving the complex zeros to the

left and away from the j u-axis as shown in Fig. (8.1.4), "10". The effect of the real

pole of the system on the system response starts to become apparent.

Increasing T- to a value greater than that in (3. a) results in moving the complex

zeros to the positions shown in Fig. (8.1.4), "11". The complex poles are dominant

and so is the real zero.

In (3. d), the complex zeros nearly coincide with the complex poles of the system

leaving the real pole and the real zero as the singularities that determine the behaviour

of the system. The response is approximately a decaying exponential.

Fig. (8.1. 5), (a), shows the case where the pair of complex poles of the system

is cancelled by the n = 1 zeros generated by the input modifier. The response after the

instant the second step is applied is determined by the effect of the real pole of the

system and the real zero of the modified input, see Fig. (8.1.4), "13". In (5. e), the

system output and the output of the first stage are repeated for the input modifica¬

tion given in (5. a). In (5.b), T. is very small, consequently the zeros generated

by the input modifier lie very far away from the origin of the complex frequency s-

plane, since -=— is now very large, and as to be expected, the response is approxi¬

mately the same as the pure step response of the system.

In (5. f), T- is very large. The zeros generated by the input modifier lie very

close to the jio-axis, further, neighbouring zeros are seperated by very short di¬

stances from one another, see Fig. (8.1.4), "15". The system response is oscilla¬

tory because the complex pole pair has not been cancelled. Keeping T1 fixed but de¬

creasing— the response in Fig. (8.1.6), (a), is obtained. The response is slightly

oscillatory because the complex poles of the system and the n = 1 pair of complex ze¬

ros of the modified input are close to each other see Fig. (8.1.4), "16". In (6.b) the

system response and the response of the first stage are shown. The behaviour of

the response in the phase plane is given in (6. c). The effect of decreasing T. and of

increasing it from its value in (6. a) on the system response may be seen from the

behaviour of the response in the phase plane for these two cases as given in (6. d) and

(6.e) respectively.

- 149 -

(a) (e)

(b)

(c)

(d)

-c(t)

Fig. (8.1. 5) Oscillograms illustrating Posicast Control for Third-Order Systems

a, b, f inputs and the corresponding outputs

e, c,g outputs of the second stage and of the third-order system

d, h the behaviour of the outputs in the phase plane. The curve in the

lower half of the phase plane represents the response of the third-

order system to a negative step

- 150

(a)

c(t)

c(t)

-*>-«.

iAiA/

(b) (e)

(c)

Fig. (8.1.6) Oscillograms illustrating Posicast Control for Third-Order Systems

a input and corresponding output

b output of the second-stage and of the third-order system

c, d, e (c): the behaviour of the output of (a) in the phase plane, (d), (e):

The effect of decreasing and of increasing T1 respectively

To obtain a deadbeat response an infinite number of ways of modifying the input

step exist ranging from theoretical ones which require the use of impulses and doublets

to others such as those shown in Fig. (8.1.7). The modified input given in (7. c) is ne-

arly'bang-bang' and consequently its response has a very short rise time. The respon¬

se shown in (7.b) was obtained by injecting a small pulse to counteract the initially ne¬

gative oscillations shown in Fig. (8.1.6), (a).

Oscillogram (7. a) is given to show that although the first step is smaller than the

required steady value it is still possible to obtain a deadbeat response. As to be expec¬

ted, the rise-time of the response in (7. a) is longer than that of either the responses

in (7.b) or (7. c). Oscillograms (7. d), (7. e) and (7.f) show the output of the third-or¬

der system and that of the first stage for the input modifications given in (7. a), (7.b)

and (7. c) respectively.

- 151 -

A third-order system having three real poles and another having a real pole and

a pair of dominant complex poles were used in similar studies. In the case of the sy¬

stem which has three real poles, it was easy to cancel any one of the real poles also

to obtain a deadbeat response. For the system having a real pole and a pair of domi¬

nant complex poles, the complex poles could be cancelled using the methods of chap¬

ter VII, and to obtain a deadbeat response it was necessary to inject a small positive

pulse at the beginning of the second step so as to counteract the effect of the real pole.

(a) (d)

(b) (e)

H**ttt J

(c) (f)

Fig. (8.1.7) A Variety of Modified Inputs which give a Deadbeat Response

a, b, c inputs and the corresponding outputs

d, e, f outputs of the second stage and of the third-order system

- 152 -

8.2. 0 Fourth-Order Systems

A fourth-order system was simulated by cascading two underdamped second-

order systems composed of RLC components. The step response of the first under-

damped second-order system is given in Fig. (8.2.1), (a), and that of the second

underdamped second-order system is given in Fig. (8.2.1), (b). The step response

of the fourth-order system is shown in Fig. (8.2.1), (c).

Oscillograms (2. a), (2.b), (2.f) and (2.g) of Fig. (8. 2.2) show that the step

response of the fourth-order system may be reduced to approximately that of either of

its constituent underdamped second-order systems by cancelling the appropriate pair

of complex poles of the fourth-order system.

On modifying the step response as shown in (2. c) a deadbeat response is obtai¬

ned, see (2. c), (2. d) and (2. h). In oscillogram (2. e) the response of the fourth-order

system to the modifying pulse used in (2. d) is given. In (2. i) the modified input used

in (2. c) and the output ot the first underdamped second-order system are shown. Os¬

cillogram (2. j) shows another choice of a modified input which gives a deadbeat respon¬

se.

8.3.0 Fifth-Order Systems

A fifth-order system was formed by cascading a first-order system having the

step response shown in Fig. (8.3.1), (a) and an oscillatory fourth-order system.

The step response of the fifth-order system is given in (l.b). The response of the

fifth-order system and that of the first-order system are repeated in oscillogram (l.c).

1 \ti

V \f\ 'V \y> >M

'1 1(a)

A>m

(b)

I"i.<Ay»tti

(c)

Fig. (8.2.1) Oscillograms of Fourth-Order System Step Responses

a the step response of the first stage alone

b the step response of the second stage alone

c the step response of the fourth-order system obtained by cascading the

stages used in (a) and (b)

- 153 -

t y^aat m

(a)

*>- c (t)

""•1\5W<«i4

(b)

mL

(g)

mi i mi .mm mi MM ****

t

) 1(c)

llri c ^ Ikl

vV5

v•*.

(h)

^ -

(d) 0)

+*+ ++y + mt mh

- ; ^_^—-

(e)

\

J1 V

-ll 5*J

(1)

Fig. (8.2.2) Oscillograms illustrating Posicast Control for a Fourth-Order System

a,b, c inputs and the corresponding outputs

f,g,h the behaviour in the phase plane of the responses given in (a), (b)

and (c) respectively

d a portion of oscillogram (c)

e the modifying pulse for case (d)

i the output of the second stage for the input shown in (d)

j a modified input and the corresponding output

154 -

<=_—^—1

(a)

-

s~~ lmm\m** >*"*"*•"•* 1J

_

(e)

_

A

"""J 7__£ t

(i)

—_

~2-—=—t~(b)

I(£)

„

P*-»«-^«

(J)

;

"!«=»—\»— 11

(c)

-3=

„2^^m i-f^ —X

-* 1

_

A1

Tr

1

/ \(g) 00

"?«- =-""3 i 1"".

17:

-

mm^jmmmmmm^J

-

~-i"*F~-X7

(d) (h) (1)

Fig. (8.3.1) Oscillograms illustrating Posicast Control for a Fifth-Order System

a step response of fifth stage alone

b step response of fifth-order system

c the responses of (a) and (b) repeated

d step response of the first four stages together and of the fifth-order

system

e input and corresponding output of fifths-order system

f, g response of (e) repeated together with those of the previous two stages

h an input and the corresponding output of the fifth-order system

i, j the response of the third and fourth stages for the input used in (h)

k, 1 a pulse of the modification (h) missing and its effect on the responses

of the fourth and fifth-order systems respectively

- 155 -

In (1. d) the step response of the fifth-order system and the input to the first-

order system are shown. By cancelling the complex poles of the fifth-order system

the response is an exponential as is shown in (1. e). The response given in (1. e) is

repeated in (l.f) and (l.g) together with that of the previous two stages, hi (l.h), it

is shown that a deadbeat response may be obtained. The behaviour of the responses

of the first two stages of the fifth-order system are given in (1. i) and (1. j) respec¬

tively for the case where the output is deadbeat. Finally shown in oscillograms (l.k)

and (1.1) respectively are the responses of the fourth-order system and of the fifth-

order system to the input used in (1. h) with the second pulse missing.

8.4.0 Conclusions

A signal composed of two steps delayed with respect to each other possesses

z-plane zeros. These zeros can be located anywhere in the entire s-plane by proper

adjustment of the magnitude of the steps and of the time delay between them. If such

a modified signal is used as the input to a system having real and/or complex poles,

then it is possible to adjust the shape of the modified input so that its zeros cancel

any real pole of the system or any pair of complex poles. Thus the step response of

a third-order system may be made to correspond to approximately the step response

of either a first or a second-order system by performing a simple modification of

the input. The same argument is also true for higher-order systems. The poles of

the system need not be cancelled by the n = 0 zeros of the modified input, they can

be cancelled by other zeros depending upon the case under consideration.

To obtain a deadbeat response many factors must be considered such as the

allowable amplitudes of the modified input, the relative location of the poles of the

system and the required rise-time. In general an infinite number of choices of a

modified input exists. To obtain the theoretically shortest rise-times, the modified

inputs must be composed of functions such as steps, impulses, doublets, etc. If the

maximum and the minimum levels of the inputs are fixed, then the responses having

the shortest rise-times may be obtained by using "bang-bang" modifiers. If the in¬

stants of application of the steps are fixed, then the methods of discrete compensa¬

tion [17] may be used to determine the input modifiers. In general, for the response

of an n -order system to be deadbeat, the zeroing of (n-1) derivatives of the error

- 156 -

is necessary. To determine the switching instants a solution of a set of differential

equations is usually required. The switching instants correspond in the phase space

to points of intersection of phase trajectories. The solution of the set of differential

equation is sometimes a source of great difficulties due to the common occurrence

of transcendental equations. In such cases, resort to graphical methods or to a di¬

gital computer aids in overcoming the difficulties. In general, in dealing with sy¬

stems of an order higher than the second, the calculations begin to become tedious,

consequently, it is felt that the best approach for most cases would be that of analog

computer simulation. If the latter method is chosen, then a signal generator of the

type described in chapter HI will prove to be very useful.

- 157 -

Chapter IX

Adaptive Posicast Control

9.0.0 Introduction

In practical automatic control situations often arise where the statistics of the

inputs and/or the process dynamics are not known or are gradually changing. In

such cases, it is necessary to be able to estimate the unknown variations during sy¬

stem operation and to adjust the controller accordingly to give satisfactory perfor¬

mance. A control system that operates in this way may be called an adaptive or

self-adjusting control system.

If the parameters of the system do not change or if they change by only small

amounts, then to design the controller the standard methods of synthesis may be

used [37], or the techniques of optimum control if optimization is important [5],

If, however, the parameters of the system change considerably, then the latter me¬

thods of synthesis will fail completely to provide a satisfactory controller and resort

to the techniques used in designing self-adjusting systems is the only solution.

Self-adjusting systems may be defined as being systems with feedback, where

the fedback information is an aid to a process of identification.

Identification is essential in self-adjusting systems. Several methods of signal

and plant identification exist. Some require the use of an external signal - that is

test signal - as the parameter perturbation technique, while others do not, such as

Kalman's procedure [23].

The controller of a self-adjusting system is never a linear system with constant

coefficients. B must be either a linear system with variable coefficients or a nonli¬

near system. A selected bibliography on self-adjusting systems is given in reference

[14]. In this chapter a method is presented for achieving adaptive posicast control.

- 158 -

9.1.0 Signal-Flow Diagram

In chapter VII, the posicast method of control was discussed with reference

to an underdamped second-order system. It was shown that if the step input to an

underdamped second-order system is modified as in Fig. (7.4.3), (b), then by choo¬

sing "T." and "—" appropriately, a deadbeat response may be achieved. The reason

for this was shown to be that the modified signal had zeros which coincided with the

poles of the underdamped system and so cancelled them.

If the poles of the underdamped system change their position in the complex

frequency s-plane, then by continuously adjusting "T," and "—" it should be still

possible to obtain a deadbeat response. In practice, however, this can never be

achieved since an instantaneous adjustment of "T.," and "—" is not possible because

time is needed for system identification. If the motion of the poles of the system is

slow compared to the time needed for system identification and adaptive controller

adjustment, then satisfactory results could be obtained.

As was mentioned, to obtain a deadbeat response it is necessary to adjust two

parameters "T-" and the ratio "—". Fortunately it was shown by Hamza [12], that

if the damping coefficient of the second-order system varied, then it is possible to

obtain satisfactory performance by adjusting only the ratio "—". The range of varia¬

tion of the relative damping in the paper referred to above was from 0.1 to 1.1. B

was recommended that "T- " should be chosen so that the imaginary component of the

n = 0 pair of zeros of the modified input are equal to those of the pair of complex

poles of the second-order system for the case of least damping expected.

The method of adaptive posicast control to be presented here and whose sig¬

nal-flow diagram will next be given employs the modified input itself as a source of

system perturbation. In other words, no external test signals are used to help identi¬

fy the system which - in many cases - is a great advantage.

A simplified signal-flow diagram of the adaptive system is given in Fig. (9.1.1).

The input is in the form of step signals. The controller is a posicast controller which

is capable of varying the ratio "—" as a function of a signal f (t). r (t) is the output

of the controller and is referred to as the modified input. e(t) is the error of the system.

c (t) its output and G (s) its transfer function. C is a computer which identifies the

motion of the poles of G (s) in the complex frequency s-plane, computes how the ratio

"—j| must be changed to achieve a deadbeat response and finally transmits the results

of the latter calculation to the controller CO via the signal f (t) to achieve the required

modification of the input.

- 159 -

»c(t)

Fig. (9.1.1) Simplified Signal-Flow Diagram of Adaptive Posicast Control System

r(t) input

modified input

c(t) output

e(t) error

f(t) output of computer

CO controller

G(s) System transfer function

C computer

From Fig. (9.1.1) it may be seen that the controller CO lies in an open loop,

which means that the system investigated is an open-loop system. The system

output does not influence the controller input, however, it influences its output,

which from the definition given in section (9.0.0) is the reason why this system is

a self-adjusting control system. Feedback from the output exists and is used in a

process of identification. The results of the identification in turn have an effect on

the system output. Due to the presence of the adaptation loop the system output -

in the presence of disturbances - will in many cases be a satisfactory one, however,

a much better performance can be achieved if the controller is placed in a closed loop.

For the conditions of achieving an invariant response, see Petrov [27],

A more detailed signal-flow diagram of the adaptive posicast control system

is given in Fig. (9.1.2). The input was chosen to be a square wave because it helps

in showing the effect of self-adjustment, Fig. (9.1.3). r-(t) is a dc signal and there¬

fore r,(t) is also a dc signal since Kg is an amplifier having a constant amplification.

Assuming for the moment that r,-(t) is also a dc signal which can change only during

the intervals (n-l)T + h- £ t 4 nT where n = 1,2,3 ... gives a signal f (t) as shown

in Fig. (9.1.3), (b). Since

- 160 -

r,(t)

r,(t)>-

r(t)>-

s(t) L^LtI^

5J»UJ _L___I

r5(t)

T-hT a'©

f(t)1%

^ntKhVi^iH

© h2 ep(t)

-c(t)

r(t)

rt(t)r2(t)r3(t)r4(t)r5(t)f(t)

'm^e(t)

c(t)

g1(b)

ep(t)q(t)

S(t)

u(t)

G2(s)w(t)

Fig. (9.1.2) Signal-Flow Diagram of Adaptive Posicast Control System

input signal and is a square wave

a dc signal

output of amplifier K_

a dc signal

output of amplifier K.

the sum of r .(t) and w(t)

output of sampler (1)

product of r (t) and f (t) and is referred to as the modified input

the error

the system output

a second-order system transfer function G«(s)

output of sampler (2)

relay output

output of amplifier K,

Kls (s+a)

K„the sum of s(t) and q(t)

a transfer function given by G0(s) = —-

i* S

output of G„(s)

- 161 -

R relay with deadzone

K„ . _ amplifiers having amplifications Kg, K4 and Kg respectively

T sampling period

(1)(2)(3) sampler number (1), (2) and (3) respectively

hj ,the duration of the sampler pulse-width, h- = T. of chapter VII

h(q) indicates that sampler (3) is closed when q(t) = 0 and open otherwise

Aj

indicates a delay of Aj units of time with respect to the clock sampler (1)

r (t) = r(t) • f(t) (9.1.1)m

therefore, r (t) will be as shown in Fig. (9.1.3), (c), which is similar to the modi¬

fied input used in chapter VH, refer back to Fig. (7.4.3), (b). For

Gi<s> j^hj (9-1-2>

and by choosing the constants "Kj" and "a" appropriately, the closed loop system

whose transfer function G(s) is given by

GJs)G<8> =

TTg^IJ(9-1-3)

may be made an underdamped second-order system.

In Fig. (9.1.3), (d) and (e) arbitrary sketches of the error and the sampled

error are given respectively. If the width of the relay deadzone for negative signals

is B. and if the negative saturation level of the relay is H„, then u (t) will be as shown

in Fig. (9.1.3), (f). G,(s) is the transfer function of an integrator and r3(t) is a posi¬

tive dc signal. Consequently r_(t) will be of the form shown in Fig. (9.1.3), (g).

From the latter figure, rg(t) is seen to be constant during the interval nT < t < nT +

hj which justifies the assumption made previously while deriving the form of f (t).

B should be noted that f (t) used during the interval 0 < t < h, resulted in an out¬

put having a large overshoot. The computer identified this fact and corrected f (t) so

that its value during the interval T < t < T + h, became smaller than during the

interval 0 < t < h«. If the new value of f (t) results in an overshoot or undershoot,

f (t) will be further corrected till finally - if the adaptive control system is well de¬

signed - satisfactory response is obtained. Naturally, the parameters of the adap-

- 162 -

r(t)A

f(tH

0 h,

e(tH

V.

ep(t)|

0 Z&i

a,

u(t)A

:*Xr

K2B, 7

bJUr5(t)n

r<(o>^_

w

KV^"

(a)

(b)

(O

(d)

(e)

(f)

(9)

~LT~

T1 T'h,

-»t

-*t

Tf

->t

"V

1 *-t

Fig. (9.1.3) A Sketch of some Signals of the Adaptive Posicast Control System

- 163 -

ting loop effect the performance of the system, consequently it is important to know

how to choose those parameters and to have an idea about their effect on the system

performance. This will be the subject of the following section.

9.2.0 Theoretical Analysis

For the systems given in Figs. (9.1.1) and (9.1. 2) the following equations may

be written

K„

(9.2.1)

(9.2.2)

equation (9.2.2) G(s) reduces to

(9.2.3)

(9.2.4)

(9.2.5)

(9.2.6)

Keeping "K," constant and varying "a" corresponds to varying the relative damping

? of the second-order system.

rm(t) = r(t) • f(t) (9.2.7)

G^s)Kl

s (s + a)

G(s)Gjfs)

"

l+Gjts)

Therefore by substituting equation (9. 2.1

G(s)Kl

-

2s + as + K.

which is of the general form

G(s)

2

0~

2 2s + 2 £<o s + CO

CO o

where

Kl = "I

and

a = 2*"o

- 164 -

e(t) for nT+hj ^ t 6 nT+hj+h2 (9.2.8)

ep(t) =

0 otherwise (9.2.9)

where n =0,1,2,3 ...

K„e„(t) when q(t) = 0 (9.2.10)* P

s(t) =

0 otherwise (9.2.11)

q(t) = J (9.2.12)

where

Hj for e (t) > Aj (9.2.13)

Jf = 0 for BjieJOiAj (9.2.14)

-H2 for e (t) < Bj (9.2.15)

A. and Bj are the positive and negative limits of the relay deadzone respectively.

H- and H, are the magnitudes of the positive and negative saturation levels of the

relay output respectively.

u(t) = q(t) + s(t) (9.2.16)

G2(s) = -f (9.2.17)

where K, is constant.,KS

t

w(t) = K3 J u(t) dt (9.2.18)

r4(t) = K4 r3(t) (9.2.19)

r5(t) = w(t) + r4(t) (9.2.20)

- 165 -

r2(t) = K5 r^t) (9.2.21)

r5(t) for nT < t 5 nT +ht (9.2.22)

f(t) =

r2(t) otherwise (9.2.23)

where n = 0,1,2 ...

Full information about a control system is given when the following are discus¬

sed:

a) The performance of the system in the transient state.

b) The steady state performance of the system, and

c) The stability of the system.

hi the following subsections these points are discussed with reference to the

adaptive control system under study.

9.2.1 The Transient and Steady State Performance of the Adaptive Control System

Suppose that the relative damping of the second-order system suddenly changes

and that previous to this change the step response of the control system was deadbeat.

In otherwords previous to the sudden change of the system damping coefficient f(t)

was correctly chosen. By readjusting f (t) a deadbeat response may once more be ob¬

tained. The quicker this adjustment occurs the better are the qualities of the adaptive

control system. To study the factors which determine the speed of adjustment of f (t)

to its required value - denoted by f (t) - it will be assumed that the duration h, is

small compared to h, and that e (t) is linearly related to e (t) where e (t) is defined

by

ex(t) = wr(t) - Wj(t) (9.2.24)

w (t) is the required value of w(t). If such assumptions are made, the signal-flow

diagram which describes the behaviour of the adaptive loop may be reduced to that

shown in Fig. (9.2.1). From the latter it is seen that w,(t) is the output of the

- 166 -

zero-order hold circuit H(s), where the input to the hold circuit are samples of

w(t).

As a start suppose that w (t) = u,(t) and that the relay is an ideal one whose

output can assume only the values t Z units. Suppose further that the amplifier K-

is disconnected so that u(t) = q(t). Now as soon as the step input is applied e(t)

jumps to a value of K u_.(t) units. At t = h.., the sampler closes causing the output

w(t) to increase according to equation (9.2.18) where u(t) = Z. The limits of integra¬

tion are h1 to h2. The magnitude of w(h..+h,) depends upon the value of K-Z-, see

Fig. (9.2. 2). After time t = (h. +h„) has elapsed, the sampler - which has a pulse

width h0 - opens and e (t) drops to a value of zero. In practice noise will be presentz p

at the input of the relay and since the latter was assumed to be ideal, it will start

to chatter resulting in an unsatisfactory behaviour of w(t), Fig. (9.2.2). By using

a relay having a deadzone, the chatter can be eliminated and the response can be

improved.

wr(t)> Qex(t)

w,(t)

Kx

e(t) © raM_Jq(tr~^J2 P

R

H(s)

(t)G,(s) *w(t)

wi[t) ©

Fig. (9.2.1) Signal-Flow Diagram of Adaptation Loop

w (t) required value of w (t)

H(s) zero-order hold circuit

w(t) the output of the integrator G„(s)

- 167 -

w(t)A

wr(t)2.

-lf-1^ -*

Fig. (9.2.2) Behaviour of w(t) when an Ideal Relay is used

w(t) output of G2(s), refer to Fig. (9.1.2)

w (t) required value of w (t)

1,2,3 represent increasing values of K«Z where t Z units are the only

possible values of the relay output'

Consider next the case where the relay has a symmetrical deadzone of width

A. For w (t) = u_-(t), e (t) behaves as shown in Fig. (9.2.3), (a). The sampling

period T is assumed to be a constant. This is not necessary, however, important

is that T must exceed a certain value to be determined later, e (t) is shown in Fig.

(9.2.3), (b). The behaviour of w(t) depends upon the value of KJ.Z. For K,Z large,

w(t) is as shown in Fig. (9. 2.3), (c) curve "1", and indicates the presence of a li¬

mit cycle, e (t) which corresponds to this case is given in (b). For smaller values

of K,Z the response denoted by "2" may be obtained. For this case the amplitudesA

of the second, third, etc. pulses of e (t) are smaller than -*• and further since A

is small the system step response c(t) will be nearly deadbeat. For case "3" KgZwas chosen to be smaller than in "2". In case "3" w(t) will keep increasing till the

amplitude of e (t) gets to be smaller than -*• whereupon the system response willP «

reach its steady state. Increasing the width of the deadzone can lead to stable states

with larger steady state errors. An unsymmetrical deadzone increases the range of

possible stable states with positive steady state errors (or vice versa).

If the relay has unequal saturation levels, that is, if H- £ H„, then instability

can easily occur. For example, if K3H, was greater than K-H, and for w (t) taken

to be a step, a possible w(t) could be the one shown in Fig. (9.2.4). It is interesting

to note that a system could be made initially unstable then finally stable.

- 168 -

ex(t)A

ep(tH

H (-

h, h*h2

H K-

*h, T+h+K, 2

H h

ZW^ZWyhj 3T

-*t

-M-Af

w(t)|

wr(t)

h, h+l^ T T'h, T^ 2T 21^ 2T*hfh2 3T

ish, h*h2 T T^h, -Rifh2 2T ZT^ 2T*h+h2 3T

Fig. (9.2.3) Behaviour of w(t) when a relay with Deadzone is used

e (t) the difference between w (t) and w«(t), see Fig. (9.2.1)

e (t) the sampled error

w(t)t the output of integrator G„(s)

w.

A

w (t) the required value of w (t)

^t

half the width of the relay deadzone when the latter is symmetri¬

cal

1,2,3 represent decreasing values of K-Z, where i Z units & 0 are the

only possible values of the relay output

- 169 -

W(t)A

Wr(t) /= ^KT T^ T'hfo 2T ZT^ 21^ 3T

Fig. (9.2.4) Behaviour of w(t) when a relay with unequal saturation levels is used

and when relay and amplifier combination is used

1 relay with unequal saturation levels used

2 relay and amplifier combination used

w(t) output of integrator G„(s), see Fig. (9.1.2)

w (t) required value of w (t)

This may be achieved by adjusting IT., H_ and the deadzone width A so that

after a few samples eft) enters the deadzone region. If the magnitude of KgH. is

chosen to be larger than that of K„H2 but not by so much that an overshoot takes

place or that e (t) enters the deadzone region after one sample, then the settling-

time when w (t) is a positive step will be shorter than when it is a negative one.

If the relay is replaced by the amplifier K„ and if K- is made very large, then

instability will occur. For medium amplification the output will tend towards its re¬

quired value after a few samples. By employing both the relay with deadzone and the

amplifier K, as shown in Fig. (9.2.1), the system becomes one which operates as a

"bang-bang" system for large signals and as a linear one for small signals. Such an

adaptive control system combines the advantages of high response speed and low

steady state error. A possible behaviour of w(t) when the relay and amplifier com¬

bination is used is shown in Fig. (9.2.4), (2). It is recommended that K, be chosen

such that when "e (t)" is slightly less than half the width of the relay deadzone one

sample be sufficient to take w(t) to its required value. Naturally, here also, if K„

is chosen to be too large oscillations will result and the occurrence of a limit cycle

is possible.

- 170 -

If the pulse duration h„ is made appreciable compared to h., the signal-flow

diagram of Fig. (9.2.1) will still be valid, however, instead of using the amplifier

K a nonlinear amplifier must be used. The exact calculations for the latter case

are difficult to perform because the amplification is rather complex since it de¬

pends upon the nonlinear function e (t) which in turn depends upon the past history

of the system.

T is fixed by the time required for the response to settle to within a small

fraction - for example 2 % - of its required steady state value. If this time is de¬

noted by T ,then T must satisfy

T * T (9.2.25)

If w (t) is not a step function, but is for example a ramp or a sinusoid, then

K, should be so adjusted that w (t) is reached after one sampling interval for the

case where Aw (t) - the change in w (t) during the interval (n-l)T+h1+h„ 6 t £ nT+h,

- is a maximum, that is equation (9.2.26) should be satisfied.

nT+h, +h„K, / q(t) dt » Aw„6

nT+h,r

(t) (9.2.26)

where n = 0,1,2 The deadzone should be adjusted to start to become effective

for any A w (t) smaller than its maximum value. The behaviour of w(t) for such

a case is illustrated in Fig. (9. 2. 5).

To determine the optimum choice of h„, A, K„ and K„ several factors must

be considered such as:

wr(tU

*wr2(t)

aw (t)I max

rj

->t

h, h*h2 T TVh^hftij 2T 2T*h,

Fig. (9.2.5) Recommended Behaviour of w(t) when w (t) varies

w (t) variation in w (t) during the interval (n-ljT+hj+h, < t < nT+h.,

- 171 -

a) How often does A wr (t) occur?

b) Are large overshoots or undershoots permissible if they occur very sel¬

dom or are they not permissible ?

c) The maximum allowable overshoot.

d) The speed of adaptation required, and

e) The maximum steady state error.

9.2.2 The Stability of the Adaptive Control System

The stability of the adaptive control system has been partly discussed in the

previous subsection. In this subsection, the main topic to be discussed will be the

effect of noise on the performance of the adaptive control system under investigation.

With reference to Fig. (9.1.2) and denoting the noise signal by n(t), if n(t) is

superimposed on either r(t) or rm(t), then an unstable response could result if n(t)

is comparable to r (t) in magnitude. The reason is that the control system is open-

loop. If n(t) is small in magnitude compared to r (t), then its effect on the system

response will be small because the output of the relay is determined by the signal

level of its input rather than by its exact shape, unless a borderline case exists.

This occurs when the magnitude of e (t) with no noise present is approximatelyA

"

equal to *-•

If the noise signal is applied to the input of G..(s) and also to the input of samp¬

ler (2), then an unsatisfactory response can occur if "n(t)" is large compared to the

width of the relay's deadzone. If "n(t)" is small compared to the relay deadzone,

the effect of the noise will be negligible due to the presence of the closed loop around

G,(s) and because the amplifier K, will tend to correct the effect of the disturbances.

Similarly if the disturbances occur within the system G1(s) their effect will die away

due to the presence of the feedback loop. If the disturbances occurred, however, wi¬

thin the system G(s) of Fig. (9.1.1), then the system response could become un¬

satisfactory if the magnitude of the distrubances is large because the system is

open loop.

Finally, if the disturbances occur at the input of the relay or of the amplifier

K„ or of the integrator, then their effect will be automatically eliminated unless

their magnitude is large.

(9.1.2)

Fig.

of

System

Control

Posicast

Adaptive

the

simulating

Circuit

Computer

Analog

(9.3

.1)

Fig.

M2J.K

- 173 -

(1) to a times ten input of integrator A05

I a sine wave generator used as the clock signal

n, HI circuits for actuating comparators MOJ, K and M2J, K respectively

IV circuit used to obtain r„(t), r (t) and to simulate G(s)it m

V circuit for actuating comparator M5J

VI circuit to replace Q05 when a sinusoidal modulation of the damping

coefficient of the system is required

9.3.0 Analog Computer Circuits

Li this section the analog computer circuits used to simulate the system of

Fig. (9.1.2) are presented. These circuits are given in Fig. (9.3.1). Referred to

by "I" in the latter figure is the sine wave generator employed to provide the clock

signal. Introducing the sine wave signal which is obtained from "I" to a limited high

gain amplifier, A03, a square wave is obtained. This square wave corresponds to

r(t) in Fig. (9.1.2). The outputs of circuits "II" and "in" actuate the comparators

MOJ, K and M2J, K which correspond in Fig. (9.1.2) to samplers (1) and (2) respec¬

tively. The circuit used to obtain r„(t), r (t) and to simulate G (s) is referred to by

"IV". Amplifier A18, the limiter number 17 and the limited high gain amplifier A37

simulate the relay with deadzone. To simulate the amplifier K_ of Fig. (9.1.2), po¬

tentiometer P39 in cascade with amplifiers A39 and A21 are employed. A25 simulates

the integrator G2(s) and -r5(t) is obtained as the output of A28.

Circuit "V" is used to actuate the comparator M5J which corresponds to sampler

"3" in Fig. (9.1.2). By connecting the output of potentiometer P06 to terminal G of

M3G instead of to potentiometer Q05, and the output of M3G to an input of A05 having

an amplification of ten, it is possible to obtain a sinusoidal modulation of the damping

coefficient of the second-order system. The circuits which replace Q05 to obtain a

sinusoidal modulation of the damping coefficient are indicated by "VI".

Q i 10o<O C •§ Id u s CO

"H,

inin

upin

iipi

iill

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- 175 -

The analog computer circuits employed in obtaining the results given in this

section are shown in Fig. (9.3.1).

Circuits used are:- I... V

01 R

10 C

11 C

13 R

al 0.85

bl -0.7

a2 0.05

b2 -0.9

^ 0.8

b3 -0.8

a4 0.12

b4 -0.12

a5 0.9

b5 -0.05

a6 0.7

b6 -0.7

"7 0.8

b„ -0.8

Q00 0.5

Q01 0.157 x 10

P10 0.55

P20 0.98

P07 0.5

Q05,Q06,Q07 0.9

P05 0.324

P06 0.072

P18 0.2

P25 0.45

P39 0.07

Q49 0.01

P28 0.2

176 -

Fig. (9.4.2) Effect of having a high Gain in the Adaptive Loop

Circuits used are:- I... V

Same as for Fig. (9.4.1) except that

P25 0.85

9.4.0 Results

In this section a sample is presented of the results obtained using the analog

computer circuit of Fig. (9.3.1).

In Fig. (9.4.1) it is shown that by choosing an appropriate value for P25 satis¬

factory response can be obtained after one sample. When the coefficient of P25 is too

large, oscillations result and the occurrence of a limit cycle is possible, Fig. (9.4. 2).

Coefficient P25 corresponds to K, in Fig. (9.1.2).

In section (9.2.1) it was mentioned that if "h," is comparable to "h.", see Fig.

(9.1.3), the performance of the adaptive control system can become unsatisfactory.

That this is true may be seen from Fig. (9.4.3). The first sample of e(t) resulted in

an increase of r,(t) I instead of in a decrease of it. Consequently the response du-

- 177 -

ring the second period of operation had a larger percentage overshoot than in the

first.

With switch "13" opened, Fig. (9.3.1), the adaptation is "bang-bang" and the

system performs as shown in Fig. (9.4.4). Only few sampling periods are required

for w(t) to become approximately equal to w (t), refer to Fig. (9.2.3), however,

the overshoot has a steady state value which depends upon the width of the relay

deadzone as previously mentioned in section (9.2.1).

With switch "13" closed, the adaptation is dual mode. The system performs

as shown in Fig. (9.4. 5). As soon as I e (t) I drops to a value smaller than the width

of the relay deadzone, switch "3" is closed, Fig. (9.2.1), allowing further impro¬

vement of system response.

To test the system performance under variable conditions, the second-order

system damping coefficient was varied sinusoidally. Starting with an incorrect

choice of r .(t), the behaviour of the system's response in the phase plane shown in

Fig. (9.4.6) was obtained. All overshoots - with the exception of the first, because

system adaptation was not yet effective - are less than 7 %.

Fig. (9.4.6) is a clear indication of the high quality of performance of the adap¬

tive control system described.

Li the following chapter, possible applications of adaptive posicast control are

discussed as well as some conclusions and suggestions for future work.

- 178 -

(a)

(b)

c(t)

^ --_

-

— -r- -~—

== "S =

= :=s- -_

_::—

—

= -

Jr —=

=--..- 3 "- =."- = =~

= = =-~

~- i. -j —

"

: -= ::= = ~=

j

-.-

- —

- - ——.—

~= = ~^.^ -' - -=- 3 ^

= —

~

- i i-i ^ = H= iE - _=~4 ,,----

-_=

:

-- =; = =-„—

= -^ = -_ =

—

=- =-

~

- =q -~ -

- ~

=E.

~ . =~ -3 ; -:~-~~^..-=l =: ^: ?=^ — —

^ _

— -

:_ r ~

^-• - __._

= = :-—

— "U ,

— -~—

=

=.--

- _j . _

= : ^." — = !=: = = —

= = =. -

: -9_- -- —

==^~ —

-— —

== .="- r= 3 3~_

--= =-

_-, --_j _=—

-j =

— '--.-.-^ -J

= ^-

. _ _=-"- _~- " -

--

~

—

~

.:= £

-"-jg- .

—

--3= "==3- -- | —

- - :\ .- —

- - —

-.=

=1A = -' iiii^--" E i-\~r i" =-j -H

i^hS~"

-"--"^ ~n -WO -= -

--- j*"^ = ?= =i

—"I "_::- 5E = =: _~ = i~

^ — ^

=. = J ^ =~

:_

= r- -^_-_-^.-. --=-- =-- ---

- -

- ^~- ^: ^ -

~

— ^

llS^Si^= ^ = = ~=: = 3=^ =

III1

1I

| i

jI£1

ii

11

=

H

I

Ilillll

= *-= —= = — =^ =

=E^==I- e(t)

-'S(t)

Fig. (9.4.3) Effect of having "h2" comparable to "hj"

Circuits used are:- I ...V

Same as for Fig. (9.4.2) except that

P06 0.036

P25 1.00

Q01 1.00

P20 0.67

179 -

C(t)A

0-

->c(t)

Fig. (9.4.4) Bang-Bang Adaptation

Circuits used are:

01 R Q00 0.50

10 C Q01 0.09

11 C P10 0.5499

13 R P20 0.9801

P07 0.5103

a..... a_ andb... •b7 Q05,Q06,Q07 0.90

same as for Fig. (9.4.1) P05 0.324

except that P06 0.036

a1 0.8 P18 0.20

bj -0.8 P25 0.45

P39 0.07

Q49 0.0101

P28 0.2165

180

C(t)

0--

c(t)

Fig. (9.4.5) System Response in the Phase Plane when Dual Mode Adaptation is

used. Bang-Bang and Linear.

Circuits used are:- I... V

Same as for Fig. (9.4. 4) except that

13 C

- 181 -

6(t)A

css(t)C(t)

Fig. (9.4.6) Behaviour of Adaptive Posicast Control System in the Phase Plane

when "\ " varies Sinusoidally

Circuits used are:- I... VI

Same as for Fig. (9.4.5) except that

12 R P06

02 C P30

Q30

Q34

0.90

0.1260

0.0020

0.1620

The origin of the system response axis was selected arbitrarily as shown

in the figure.

- 182 -

Chapter X

Concluding Remarks

10. 0.0 Some Remarks and Suggestions for Future Work

In chapter II the conditions under which input signals to a linear system are

dynamically equivalent - referred to as the Principle of Equivalent Areas - were

stated and derived. Expressions for error in system response were given when a

signal is used instead of another as input to a linear system and it was shown that

if a signal can be chosen arbitrarily within the period T-, refer to section (2.3.0),

then it is possible to determine a location for which the output error at the end of

this period is zero. A study was made using various inputs to an arbitrary linear

system and indications of the error that results were given. Dynamic equivalency

was used in the past in the study of pulse-width modulated and nonlinear systems.

It is useful as a means of obtaining simplified solutions to some complex problems

and should not be underestimated.

The results reported in chapter IV indicate that dual mode control should be

seriously considered as a means of improving the response of control systems. In

the study performed step inputs were used, however, improvement of system res¬

ponse for other types of inputs should be also possible.

Nonlinear damping has been known to improve the performance of control sy¬

stems. In chapter V, this method was developed further and signal adaptive samp¬

ling was introduced and its possibilities were extended. Studies similar to those

given in chapter V for higher-order systems and for inputs other than steps would

be very useful and so are recommended for future work.

In chapters VI, VII and VHI, the posicast method of control was discussed and

developed. For the large majority of cases determination of the exact form of the

modified input is very difficult, impossible - up to the present - or requires a

great amount of labour, so that analog computer simulation or the use of a signal

generator such as the variable-steps signal generator described in chapter HI are

to be recommended.

The posicast controllers employed in chapters VI to Vm were all located in

an open loop. Methods for designing simple posicast controllers which are to be plac¬

ed in the closed loop would be very useful indeed.

- 183 -

In chapter IX a method for achieving adaptive posicast control was described

and was shown to be very effective. Adaptive posicast controllers may be used as

controllers for systems having variable parameters or as automatic identifiers for

determining the locations and movement of poles of control systems. Suggestions for

future work are to extend this method to higher-order systems - for which chapter

Vm would be very useful - and to employ other types of inputs and further to de¬

sign control systems in which the adaptive posicast controller described is used as

an automatic identifier.

- 184 -

Appendix A

c (t) the unit step response of an underdamped secondrorder

system is given for various values of the relative damping.

The natural frequency of the system was CO =0.5 rad. /sec.

c.(t) the unit impulse response of an underdamped second-

order system having If = 0.2 and CO = 0. 5 rad./sec.

Some values of c.(t) near its first peak are given for various

values of t.

t = relative damping of second-order system,

t. = instant at which the unit step response c (t) of second-

order system is equal to unity for the first time

t„ = instant at which c (t) is equal to unity for the second time

t, = instant at which c (t) is equal to unity for the third time

T5-

= the half-period of the second-order system responseT

and which is given by -k- = (t2-tj) sees. The natural fre¬

quency of the circuit was Co = 0. 5 rad./sec.

- 185

% = 0.1

seconds

.0000000 .0000000

.1000000 .0012436

.2000000 .0049610

.3000000 .0111150

.4000000 .0196690

.5000000 .0305730

.6000000 .0437820

.7000000 .0592360

.8000000 .0768770

.9000000 .0966390

1.0000000 .1184500

1.1000000 .1422300

1.2000000 .1679200

1.3000000 .1954200

1.4000000 .2246400

1.5000000 .2555000

1.6000000 .2879100

1.7000000 .3217600

1.8000000 .3569700

1.9000000 .3934200

2.0000000 .4310200

2.1000000 .4696600

2.2000000 .5092400

2.3000000 .5496400

2.4000000 .5907600

2.5000000 .6324900

2.6000000 .6747100

2.7000000 .7173300

2.8000000 .7602200

2.9000000 .8032900

3.0000000 .8464100

3.1000000 .8894900

3.2000000 .9324100

3.3000000 .9750800

3.4000000 1.0174000

3.5000000 1.0592000

3.6000000 1.1005000

3.7000000 1.1411000

3.8000000 1.1810000

3.9000000 1.2200000

4.0000000 1.2581000

4.1000000 1.2951000

4.2000000 1.3311000

4.3000000 1.3658000

4.4000000 1.3994000

4.5000000 1.4315000

4.6000000 1.4623000

4.7000000 1.4917000

4.8000000 1.5195000

4.9000000 1.5458000

5.0000000 1.5704000

5.1000000 1.5934000

A 2

t

secondsc(t)

5.2000000 1.6146000

5.3000000 1.6342000

5.4000000 1.6519000

5.5000000 1.6679000

5.6000000 1.6820000

5.7000000 1.6943000

5.8000000 1.7048000

5.8999000 1.7134000

5.9999000 1.7201000

6.0999000 1.7250000

6.1999000 1.7280000

6.2999000 1.7292000

6.3999000 1.7286000

6.4999000 1.7261000

6.5999000 1.7219000

6.6999000 1.7159000

6.7999000 1.7082000

6.8999000 1.6989000

6.9999000 1.68780007.0999000 1.6752000

7.1999000 1.6610000

7. 2999000 1.6453000

7.3999000 1.6282000

7.4999000 1.6097000

7. 5999000 1.58980007.6999000 1.5687000

7. 7999000 1.5464000

7.8999000 1.5229000

7. 9999000 1.4984000

8.0999000 1.4728000

8.1999000 1.4464000

8.2999000 1.4191000

8.3999000 1.39100008.4999000 1.36220008.5999000 1.3328000

8.7000000 1.3029000

8.8000000 1.2726000

8.9000000 1.2418000

9.0000000 1.2108000

9.1000000 1.1795000

9.2000000 1.1481000

9.3000000 1.1167000

9.4000000 1.0852000

9.5000000 1.0539000

9.6000000 1.0228000

9.7000000 .9918700

9.8000000 .96131009.9000000 .9311400

$ =0.15

t

seconds

.0000000

.1000000

. 2000000

.3000000

. 4000000

.5000000

.6000000

.7000000

.8000000

.9000000

1.0000000

1.1000000

1.2000000

1.3000000

1.4000000

1.5000000

1.6000000

1.70000001.8000000

1.9000000

2.0000000

2.1000000

2.2000000

2.3000000

2.4000000

2.5000000

2.6000000

2.7000000

2.8000000

2. 9000000

3.0000000

3.1000000

3.2000000

3.3000000

3.4000000

3.5000000

3. 6000000

3.7000000

3.8000000

3.9000000

4.0000000

4.1000000

4.2000000

4.3000000

4.4000000

4. 5000000

4. 6000000

4.7000000

4.8000000

4.9000000

5.0000000

- 186

C(t)

.0000009

.0012436

.0049458

.0110620

.0195410

.0303240

.0433530

.0585620

.0758790

.0952320

.1165400

.1397200

.1647000

.1913700

.2196500

.2494500

.2806600

.3132000

.3469500

.3818200

.4177100

.4545100

.4921100

.5304100

.5693100

.6086900

.6484600

.6885100

.7287400

. 7690400

.8093100

.8494600

.8893800

.9289800

.9681700

1.0069000

1.04490001.0824000

1.11900001.1548000

1.1897000

1.22360001.2565000

1.2882000

1.31870001.3480000

1.3760000

1.40260001.4279000

1.4517000

1.4740000

t

seconds

5.1000000

5.2000000

5.3000000

5.4000000

5. 5000000

5.6000000

5.7000000

5.8000000

5.8999000

5.9999000

6. 0999000

6.1999000

6.2999000

6.3999000

6.4999000

6. 5999000

6.6999000

6.7999000

6. 8999000

6.9999000

7. 0999000

7.1999000

7.2999000

7.3999000

7.4999000

7. 5999000

7.6999000

7. 7999000

7.8999000

7. 9999000

8.0999000

8.1999000

8.2999000

8.3999000

8.4999000

8.5999000

8.7000000

8.8000000

8.9000000

9.0000000

9.1000000

9.2000000

9.3000000

9.4000000

9.5000000

9.6000000

9.7000000

9.8000000

9.9000000

C(t)

1.4948000

1.5141000

1. 5318000

1.5480000

1.5625000

1. 5754000

1.5867000

1. 5964000

1.6045000

1.6109000

1.6158000

1.6190000

1.6206000

1.6207000

1.6193000

1.6163000

1.6118000

1.6059000

1.5986000

1.5899000

1.5793000

1.5685000

1.5560000

1.5422000

1.5273000

1.5113000

1.4943000

1.4763000

1.4574000

1.4377000

1.4171000

1.3958000

1.3739000

1.3514000

1.3283000

1.3047000

1.2808000

1.2565000

1.2319000

1.2071000

1.1822000

1.1572000

1.1322000

1.1072000

1.0823000

1.0576000

1.0331000

1.0089000

.9850600

- 187 -

$ =0.2 A 4

t

secondsc(t)

t

seconds c(t)

. 0000000 .0000000 5.2000000 1.4247000

. 1000000 .0012407 5.3000000 1.4410000

. 2000000 .0049286 5.4000000 1.4558000

.3000000. 0110060 5.5000000 1.4693000

.4000000 .0194100 5.6000000 1.4813000

. 5000000 .0300750 5.7000000 1.4919000

.6000000 .0429300 5.8000000 1.5011000

. 7000000 .0578980 5.8999000 1.5088000

. 8000000 .0749000 5.9999000 1.5151000

. 9000000 .0938550 6.C999000 1.52010001.0000000 .1146800 6.1999000 1. 52360001.1000000 .1372800 6.2999G00 1.52580001.2000000 .1615700 6.3999000 1.52660001.3000000 .1874500 6.4999000 1.52610001.4000000 .2148300 6.5999000 1.52430001.5000000 .2436100 6.6999000 1.52130001.6000000 .2736900 6.7999000 1.51700001.7000000 .3049800 6.8999000 1.51160001.8000000 .3373700 6.9999000 1.50490001.9000000 .3707500 7.0999000 1.49720002.0000000 .4050300 7.1999000 1.48840002.1000000 .4401000 7.2999000 1.47850002.2000000 .4758700 7.3999000 1.46770002.3000000 .5122200 7.4999000 1.45590002.4000000 .5490600 7. 5999000 1.44320002.5000000 .5862900 7. 6999000 1.42970002.6000000 .6238000 7. 7999000 1.41540002.7000000 .6615000 7.8999000 1.40030002.8000000 .6992900 7.9999000 1.38460002.9000000 .7370800 8.0999000 1.36820003.0000000 .7747700 8.1999000 1.35120003.1000000 .8122700 8.2999000 1.33370003.2000000 .8495000 8.3999000 1.31570003.3000000 .8863600 8.4999000 1.29720003.4000000 .9227700 8.5999000 1.27840003.5000000 .9586500 8.7000000 1.259300(13.6000000 .9939200 8.8000000 1.24000003.7000000 1.0285000 8.9000000 1.22040003.8000000 1.0623000 9.0000000 1.20060003.9000000 1.0954000 9.1000000 1.18080004.0000000 1.1275000 9.2000000 1.16090004.1000000 1.1586000 9.3000000 1.14100004.2000000 1.1888000 9:4000000 1.12110004.3000000 1.2179000 9.5000000 1.10140004.4000000 1.2459000 9.6000000 1.08170004.5000000 1.2727000 9.7000000 1.06230004.6000000 1.2983000 9.8000000 1.04310004.7000000 1.3227000 9.9000000 1.02420004.8000000 1.34570004.9000000 1.36750005.0000000 1.38800005.1000000 1.4070000

- 188 -

Ij = 0.25A 6

t

secondsc(t)

t

secondsc(t)

.0000000 .0000009 6.0999000 1.4357000

.1000000 .0012388 6.1999000 1.4396000

.2000000 .0049133 6.2999000 1.4423000

.3000000 .0109530 6.3999000 1.4439000

.4000000 .0192840 6.4999000 1.4443000

. 5000000 . 0298310 6.5999000 1.4437000

.6000000 .0425130 6.6999000 1.4419000

.7000000 .0572440 6.7999000 1.4391000

.8000000 .0739390 6.8999000 1.4353000

.9000000 .0925090 6.9999000 1.4305000

1.0000000 .1128600 7.0999000 1.4248000

1.1000000 .1349000 7.1999000 1.4182000

1.2000000 .1585300 7.2999000 1.4106000

1.3000000 .1836500 7.3999000 1.4023000

1.4000000 .2101700 7.4999000 1.3932000

1.5000000 .2379800 7. 5999000 1.3833000

1.6000000 .2669900 7.6999000 1.3727000

1.7000000 . 2970900 7. 7999000 1.3615000

1.8000000 .3281800 7. 8999000 1.3497000

1.9000000 .3601700 7.9999000 1.3372000

2.0000000 .3929400 8.0999000 1.3243000

2.1000000 .4264000 8.1999000 1.3109000

2.2000000 . 4604500 8.2999000 1.2970000

2.3000000 .4950000 8.3999000 1.2828000

2.4000000 .5299300 8.4999000 1.2682000

2.5000000 .5651700 8.5999000 1.2533000

2.6000000 .6006000 8.7000000 1.2381000

2.7000000 .6361500 8.8000000 1.2227000

2.8000000 .6717200 8.9000000 1.2072000

2.9000000 .7072200 9.0000000 1.1915000

3.0000000 . 7425700 9.1000000 1.1758000

3.1000000 .7776800 9.2000000 1.1600000

3.2000000 .8124700 9.3000000 1.1442000

3.3000000 .8468700 9.4000000 1.1284000

3.4000000 .8807900 9. 5000000 1.1127000

3.5000000 .9141700 9.6000000 1.0971000

3.6000000 .9469400 9.7000000 1.0817000

3.7000000 .9790300 9.8000000 1.0664000

3.8000000 1. 0104000 9.9000000 1.0514000

3.9000000 1.0409000 10.0000000 1.0366000

4.0000000 1.0706000 10.1000000 1.0220000

4.1000000 1.0994000 10.2000000 1.0078000

4.2000000 1.1273000 10.3000000 .9939000

4.3000000 1.1541000 10.4000000 .9803600

4.4000000 1.1799000 10.5000000 .9672100

4.5000000 1.2046000

4.6000000 1.2282000

4.7000000 1.2506000

4.8000000 1.2719000

4.9000000 1.2920000

5.0000000 1.3108000

5.1000000 1.3285000

5.2000000 1.3448000

5.3000000 1.3600000

5.4000000 1.3738000

5.5000000 1.3864000

5.6000000 1.3978000

5.7000000 1.4078000

5.8000000 1.4166000

5.8999000 1.4242000

5.9999000 1.4306000

- 189 -

* = 0-3 A 5

t

secondsc(t)

t

secondsC(t)

.0000000 .0000000 6.0999000 1.3608000

.1000000 .0012360 6.1999000 1.3651000

.2000000 .0048962 6.2999000 1.3684000

.3000000 .0108990 6.3999000 1.3707000

.4000000 .0191570 6.4999000 1.3720000

.5000000 .0295890 6.5999000 1.3723000

.6000000 .0421020 6.6999000 1.3717000

. 7000000 .0566010 6.7999000 1.3703000

.8000000 .

0729970 6.8999000 1.3679000

.9000000 .0911910 6.9999000 1.3647000

1.0000000 .1110800 7.0999000 1.3607000

1.1000000 .1325800 7.1999000 1.3560000

1.2000000 . 1555700 7.2999000 1.3505000

1.3000000 .1799700 7.3999000 1.3443000

1.4000000 .2056600 7.4999000 1.3374000

1.5000000 .2325500 7.5999000 1.3299000

1.6000000 .2605400 7.6999000 1.3219000

1.7000000 . 2895200 7.7999000 1.3132000

1.8000000 .3193900 7.8999000 1.3041000

1.9000000 .3500500 7.9999000 1.2944000

2.0000000 .3814100 8.0999000 1.2844000

2.1000000 .4133700 8.1999000 1.2739000

2.2000000 .4458200 8.2999000 1.2631000

2.3000000 .4786800 8.3999000 1.2519000

2.4000000 .5118500 8.4999000 1.2404000

2.5000000 .5452500 8.5999000 1.2287000

2.6000000 .5787700 8.7000000 1.2168000

2.7000000 .6123500 8.8000000 1.2047000

2.8000000 .6458800 8.9000000 1.1925000

2.9000000 .6793000 9.0000000 1.1801000

3.0000000 .7165200 9.1000000 1.1676000

3.1000000 . 7454600 9.2000000 1.1552000

3.2000000 .7780600 9.3000000 1.1427000

3.3000000 .8102400 9.4000000 1.1302000

3.4000000 .8419400 9.5000000 1.1177000

3.5000000 .8730900 9.6000000 1.1054000

3.6000000 .9036300 9.7000000 1.0931000

3.7000000 .9335100 9.8000000 1.0810000

3.8000000 .9626600 9.9000000 1.0690000

3.9000000 .9910500 10.0000000 1.0573000

4.0000000 1.0186000 10.1000000 1.0457000

4.1000000 1.0453000 10.2000000 1.0344000

4.2000000 1.0711000 10.3000000 1.0233000

4.3000000 1.0960000 10.4000000 1.0125000

4.4000000 1.1199000 10.5000000 1.0019000

4.5000000 1.1428000 10.6000000 .9917000

4.6000000 1.1647000 10.7000000 .9818000

4.7000000 1.1855000 10.8000000 .9722400

4.8000000 1.2052000 10.9000000 .9630300

4.9000000 1.2239000 11.0000000 .9541800

5.0000000 1.2414000 11.1000000 .9457000

5.1000000 1.2579000 11.2000000 .9376100

5.2000000 1.2732000 11.3000000 . 9299200

5.3000000 1.2874000 11.4000000 .9226200

5.4000000 1.3005000 11.5000000 .9157300

5.5000000 1.3124000 11.6000000 .9092500

5.6000000 1.3232000

5.7000000 1.3329000

5.8000000 1.3415000

5.8999000 1.3490000

5.9999000 1.3554000

- 190 -

5 = 0.35 A 7

t

secondseft)

t

secondseft)

.0000000 .0000009 6.0999000 1.2940000

. 1000000 .0012350 6.1999000 1.2987000

. 2000000 .0048800 6.2999000 1.3025000

.3000000 .0108450 6.3999000 1.3054000

.4000000 .0190330 6.4999000 1.3075000

. 5000000 .0293490 6.5999000 1.3087000

. 6000000 .0416970 6.6999000 1.3092000

.7000000 .0559730 6.7999000 1.3089000

.8000000 .0720750 6.8999000 1.3078000

. 9000000 .0899040 6.9999000 1.3060000

1.0000000 .1093500 7.0999000 1.3035000

1.1000000 .1303200 7.1999000 1.3004000

1.2000000 .1527100 7.2999000 1.2966000

1.3000000 .1764000 7.3999000 1.2922000

1.4000000 .2013100 7.4999000 1.2873000

1.5000000 .2273200 7. 5999000 1.2818000

1.6000000 .2543300 7.6999000 1.2758000

1.7000000 .2822400 7. 7999000 1.2694000

1.8000000 .3109600 7.8999000 1.2625000

1.9000000 .3403800 7.9999000 1.2552000

2.0000000 .3704100 8.0999000 1.2475000

2.1000000 .4009600 8.1999000 1.2395000

2.2000000 .4319200 8.2999000 1.2312000

2.3000000 .4632200 8.3999000 1.2226000

2.4000000 .4947500 8.4999000 1.2137000

2.5000000 .5264400 8.5999000 1.2046000

2.6000000 .5582100 8.7000000 1.1954000

2.7000000 . 5899700 8.8000000 1.1859000

2.8000000 .6216400 8.9000000 1.1764000

2.9000000 .6531600 9.0000000 1.1667000

3.0000000 . 6844400 9.1000000 1.1569000

3.1000000 .7154200 9.2000000 1.1471000

3.2000000 .7460300 9.3000000 1.1373000

3.3000000 .7762200 9.4000000 1.1275000

3.4000000 .8059100 9.5000000 1.1177000

3.5000000 .8350700 9.6000000 1.1080000

3.6000000 .8636200 9.7000000 1.0983000

3.7000000 .8915300 9.8000000 1.0887000

3.8000000 .9187400 9.9000000 1.0792000

3.9000000 .9452200 10.0000000 1.0699000

4.0000000 .9709200 10.1000000 1.0607000

4.1000000 .9958100 10.2000000 1.0517000

4.2000000 1.0199000 10.3000000 1.0428000

4.3000000 1.0430000 10.4000000 1.0342000

4.4000000 1.0653000 10. 5000000 1.0257000

4.5000000 1.0866000 10.6000000 1.0175000

4.6000000 1.1070000 10.7000000 1.0096000

4.7000000 1.1264000 10.8000000 1.0019000

4.8000000 1.1449000 10.9000000 .9944000

4.9000000 1.1623000 11.0000000 .9872200

5.0000000 1.1788000 11.1000000 .9803100

5.1000000 1.1943000 11.2000000 .9736900

5.2000000 1.2087000 11.3000000 .9673600

5.3000000 1.2221000 11.4000000 .9613300

5.4000000 1.2346000 11.5000000 .9556100

5.5000000 1.2460000 11.6000000 .9501900

5.6000000 1.2564000 11.7000000 .9450700

5.7000000 1.2659000 11.8000000 .9402700

5.8000000 1.2743000 11.9000000

5.8999000 1.2818000

5.9999000 1.2884000

- 191 -

% = 0.4 A 8

t

secondsc(t)

t

secondsc(t)

.0000000 .0000038 6.1999000 1.2391000

.1000000 .0012369 6.2999000 1.2433000

.2000000 .0048676 6.3999000 1.2469000

.3000000 .0107960 6.4999000 1.2496000

.4000000 .0189140 6.5999000 1.2517000

.5000000 .0291190 6.6999000 1.2530000

.6000000 .0413010 6.7999000 1.2537000

.7000000 .0553550 6.8999000 1.2538000

.8000000 .0711730 6.9999000 1.2532000

.9000000 .0886460 7.0999000 1.2520000

1.0000000 .1076700 7.1999000 1.2502000

1.1000000 .1281300 7.2999000 1.2479000

1.2000000 .1499200 7.3999000 1.2451000

1.3000000 .1729500 7.4999000 1.2418000

1.4000000 .1970900 7.5999000 1.2380000

1.5000000 .2222600 7.6999000 1.2338000

1.6000000 .2483500 7.7999000 1.2292000

1.7000000 .2752500 7.8999000 1.2242000

1.8000000 .3028800 7.9999000 1.2189000

1.9000000 .3311300 8.0999000 1.2132000

2.0000000 .3599100 8.1999000 1.2072000

2.1000000 .3891300 8.2999000 1.2010000

2.2000000 .4187000 8.3999000 1.1945000

2.3000000 .4485000 8.4999000 1.1878000

2.4000000 .4785600 8.5999000 1.1809000

2.5000000 .5086800 8.7000000 1.1738000

2.6000000 .5388200 8.8000000 1.1665000

2.7000000 .5689100 8.9000000 1.1592000

2.8000000 .5988800 9.0000000 1.1517000

2.9000000 .6286500 9.1000000 1.1442000

3.0000000 .6581700 9.2000000 1.1366000

3.1000000 .6873600 9.3000000 1.1290000

3.2000000 . 7161800 9.4000000 1.1213000

3.3000000 . 7445600 9.5000000 1.1137000

3.4000000 .7724600 9.6000000 1.1060000

3.5000000 .7998200 9.7000000 1.0984000

3.6000000 .8265900 9.8000000 1.0909000

3.7000000 .8527500 9.9000000 1.0835000

3.8000000 .8782400 10.0000000 1.0761000

3.9000000 .9030200 10.1000000 1.0688000

4.0000000 .9270800 10.2000000 1.0617000

4.1000000 .9503600 10.3000000 1.0546000

4.2000000 .9728600 10.4000000 1.0478000

4.3000000 .9945400 10.5000000 1.0410000

4.4000000 1.0154000 10.6000000 1.0345000

4.5000000 1.0354000 10.7000000 1.0281000

4.6000000 1.0545000 10.8000000 1.0219000

4.7000000 1.0727000 10.9000000 1.0158000

4.8000000 1.0901000 11.0000000 1.0100000

4.9000000 1.1065000 11.1000000 1.0044000

5.0000000 1.1220000 11.2000000 .9990000

5.1000000 1.1367000 11.3000000 .9938000

5.2000000 1.1504000 11.4000000 .9888300

5.3000000 1.1632000 11.5000000 .9840700

5.4000000 1.1751000

5.5000000 1.1861000

5.6000000 1.1963000

5.7000000 1.2055000

5.8000000 1.2139000

5.8999000 1.2214000

5.9999000 1.2281000

6.0999000 1.2340000

- 192 -

5 = 0.45 A 9

t

secondsc(t)

t

secondsc(t)

.0000000 .0000028 6.4999000 1.1974000

.1000000 .0012331 6.5999000 1.2002000

.2000000 .0048513 6.6999000 1.2023000

.3000000 . 0107410 6.7999000 1.2039000

.4000000 .0187900 6.8999000 1.2049000

.5000000 .0288830 6.9999000 1.2053000

.6000000 .0409070 7.0999000 1.2052000

.7000000 . 0547450 7.1999000 1.2047000

.8000000 .0702830 7.2999000 1.2036000

.9000000 .0874100 7.3999000 1.2021000

1.0000000 .1060200 7.4999000 1.2002000

1.1000000 . 1259900 7.5999000 1.1979000

1.2000000 . 1472100 7.6999000 1.19520001.3000000 . 1695900 7.7999000 1.1921000

1.4000000 .1930100 7.8999000 1.1887000

1.5000000 .2173800 7.9999000 1.1850000

1.6000000 . 2425900 8.0999000 1.18100001.7000000 .2685300 8.1999000 1.1767000

1.8000000 .2951300 8.2999000 1.1722000

1.9000000 .3222700 8.3999000 1.16750002.0000000 .3498800 8.4999000 1.16250002.1000000 .3778600 8.5999000 1.15740002.2000000 .4061300 8.7000000 1.15210002.3000000 .4346100 8.8000000 1.14670002.4000000 .4632200 8.9000000 1.14120002.5000000 .4918800 9.0000000 1.13550002.6000000

. 5205200 9.1000000 1.12980002. 7000000 . 5490700 9.2000000 1.12400002.8000000

. 5774700 9.3000000 1.1182000

2.9000000 . 6056500 9.4000000 1.1123000

3.0000000 . 6335600 9.5000000 1.10640003.1000000 .6611300 9.6000000 1.10060003.2000000 .6883200 9.7000000 1.0947000

3.3000000 .7150700 9.8000000 1.08880003.4000000 .7413400 9.9000000 1.08300003.5000000 .7670900 10.0000000 1.07730003.6000000

. 7922800 10.1000000 1.07160003.7000000 .8168600 10.2000000 1.06600003.8000000 .8408100 10.3000000 1.06050003.9000000 .8641000 10.4000000 1.05500004.0000000 .8866900 10.5000000 1.04970004.1000000 .9085700 10.6000000 1.04450004.2000000 . 9297000 10.7000000 1.03940004.3000000 . 9500800 10.8000000 1.03440004.4000000 .9696800 10.9000000 1.02960004. 5000000 .9885000 11.0000000 1.02490004.6000000 1.0065000 11.1000000 1.02040004.7000000 1.0237000 11.2000000 1.01600004.8000000 1.040WOO 11.3000000 1.01170004.9000000 1.0557000 11.4000000 1.00760005.0000000 1.0704000 11.5000000 1.00370005.1000000 1.0844000 11.6000000 .99996005.2000000 1.0975000 11.7000000 .99637005.3000000 1.1098000 11.8000000 .99294005.4000000 1.1213000 11.9000000 .98969005.5000000 1.1319000 12.0000000 .98660005.6000000 1.1418000 12.1000000 .98368005.7000000 1.1509000 12.2000000 .98092005.8000000 1.15930005.8999000 1.1669000

5.9999000 1.1737000

6.0999000 1.17980006.1999000 1.1852000

6.2999000 1.19000006.3999000 1.1940000

- 193 -

t = 0.50 A 10

t

secondsc(t)

t

secondsc(t)

.0000000 .0000057 6.3999000 1.1460000

.1000000 .0012341 6.4999000 1.1500000

.2000000 .0048370 6.5999000 1.1533000

.3000000 . 0106920 6.6999000 1.1562000

.4000000 .0186720 6.7999000 1.1585000

.5000000 .0286540 6.8999000 1.1603000

. 6000000 0405210 6.9999000 1.1616000

. 7000000 .0541470 7.0999000 1.1625000

.8000000 .0694140 7.1999000 1.1630000

.9000000 .0862050 7.2999000 1.1630000

1.0000000 .1044100 7.3999000 1.1626000

1.1000000 .1239000 7.4999000 1.1619000

1.2000000 .1445800 7.5999000 1.1607000

1.3000000 .1663400 7.6999000 1.1593000

1.4000000 . 1890700 7.7999000 1.1575000

1.5000000 . 2126700 7.8999000 1.15550001.6000000

. 2370400 7.9999000 1.15310001.7000000 . 2620700 8.0999000 1.1505000

1.8000000 .2876900 8.1999000 1.1477000

1.9000000 .3137900 8.2999000 1.14460002.0000000 .3403000 8.3999000 1.14130002.1000000 .3671100 8.4999000 1.1379000

2.2000000 .3941700 8.5999000 1.13430002.3000000 .4213800 8.7000000 1.13050002.4000000 .4486800 8.8000000 1.1266000

2.5000000 .4759800 8.9000000 1.1226000

2.6000000 .5032400 9.0000000 1.11840002.7000000 . 5303700 9.1000000 1.11420002.8000000 .5573300 9.2000000 1.10990002.9000000 . 5840500 9.3000000 1.1056000

3.0000000 .6104900 9.4000000 1.1012000

3.1000000 .6365800 9.5000000 1.09680003.2000000 .6622800 9.6000000 1.0923000

3.3000000 . 6875600 9.7000000 1.08790003.4000000 .7123600 9.8000000 1.08340003.5000000 .7366600 9.9000000 1.07900003.6000000 .7604100 10. 0000000 1.0746000

3.7000000 . 7835900 10.1000000 1.07020003.8000000 .8061700 10.2000000 1.06590003.9000000 .8281200 10.3000000 1.06160004.0000000 .8494200 10.4000000 1.05740004.1000000 .8700400 10.5000000 1.05320004.2000000 .8899800 10.6000000 1.04910004.3000000

.9092100 10.7000000 1.0451000

4.4000000 .9277200 10.8000000 1.04120004.5000000 .9455100 10.9000000 1.03740004.6000000 .9625700 11.0000000 1.03360004.7000000 .9788800 11.1000000 1.03000004.8000000 .9944500 11.2000000 1.02640004.9000000 1.0093000 11.3000000 1.02300005.0000000 1.0233000 11.4000000 1.01970005.1000000 1.0367000 11.5000000 1.01650005.2000000 1.0493000 11.6000000 1.01340005.3000000 1.0611000 11.7000000 1.01050005.4000000 1.0723000 11.8000000 1.00760005.5000000 1.0827000 11.9000000 1.00490005.6000000 1.0924000 12.0000000 1.00230005.7000000 1.1014000 12.1000000 .99980005.8000000 1.1097000 12.2000000 .99744005.8999000 1.1174000

5.9999000 1.1243000

6.0999000 1.1307000

6.1999000 1.13640006.2999000

11.1415000

- 194 -

$ = 0.55 A 11

secondsc(t)

t

secondsC(t)

.0000000 .0000057 7.2999000 1.1255000

.1000000 .0012302 7.3999000 1.1261000

.2000000 .0048218 7.4999000 1.1263000

.3000000 .0106380 7.5999000 1.1262000

.4000000 .0185530 7.6999000 1.1258000

.5000000 .0284290 7.7999000 1.1252000

.6000000 .0401390 7.8999000 1.1242000

.7000000 .0535580 7.9999000 1.1230000

.8000000 . 0685600 8.0999000 1.1216000

.9000000 .0850240 8.1999000 1.1200000

1.0000000 . 1028300 8.2999000 1.1181000

1.1000000 .1218700 8.3999000 1.1161000

1.2000000 . 1420200 8.4999000 1.1139000

1.3000000 .1631900 6.5999000 1.1115000

1.4000000 . 1852500 8. 7000000 1.1090000

1.5000000 .2081200 8.8000000 1.1063000

1.6000000 .2316900 8.9000000 1.1036000

1.7000000 . 2558600 9.0000000 1.1007000

1.8000000 .2805000 9.1000000 1.0978000

1.9000000 .3056700 9.2000000 1.0947000

2.0000000 .3311400 9.3000000 1.0916000

2.1000000 .3568600 9.4000000 1.0885000

2.2000000 .3827800 9.5000000 1.0852000

2.3000000 .4088100 9.6000000 1.0820000

2.4000000 . 4348800 9.7000000 1.0787000

2.5000000 .4609300 9.8000000 1.0754000

2.6000000 .4869000 9.9000000 1.0722000

2.7000000 .5127300 10.0000000 J.0689000

2.8000000 .5383600 10.1000000 1.0656000

2.9000000 .5637400 10.2000000 1.0623000

3.0000000 .5888300 10.3000000 1.0591000

3.1000000 .6135700 10.4000000 1.0559000

3.2000000 .6379300 10.5000000 1.0527000

3.3000000 .6618600 10.6000000 1.0495000

3.4000000 .6853400 10. 7000000 1.0464000

3.5000000 . 7083200 10.8000000 1.0434000

3.6000000 . 7307800 10.9000000 1.0404000

3.7000000 . 7527000 11.0000000 1.0375000

3.8000000 . 7740500 11.1000000 1.0347000

3.9000000 . 7948000 11.2000000 1.0319000

4.0000000 .8149400 11.3000000 1.0292000

4.1000000 .8344600 11.4000000 1.0265000

4.2000000 .8533300 11.5000000 1.0240000

4.3000000 .8715500 11.6000000 1.0215000

4,4000000 .8891000 11.7000000 1.01910004.5000000 . 9059900 11.8000000 1.0168000

4.6000000 .9222000 11.9000000 1.0145000

4.7000000 .9377300 12.0000000 1.0124000

4.8000000 .9525900 12.1000000 1.0103000

4.9000000 .9667600 12.2000000 1.00830005.0000000 . 9802600 12.3000000 1.00640005.1000000 . 9930800 12.4000000 1.00460005.2000000 1.0052000 12.5000000 1.00290005.3000000 1.0167000 12.6000000 1.00120005.4000000 1.0276000 12. 7000000 .99969005.5000000 1.0377000 12.8000000 .9982200

5.6000000 1.0473000 12.9000000. 9968300

5.7000000 1.0562000 13.0000000. 9955200

5.8000000 1.0645000 13.1000000 .99430005.6999000 1.0722000 13.2000000 .9931500

5.9999000 1.0793000 13.3000000 .99208006.0999000 1.0856000 13.4000000

. 99109006.1999000 1.0919000 13.5000000

. 9901800

6.2999000 1.0973000 13.6000000 .98933006.3999000 1.1022000 13.7000000

. 98856006.4999000 1.1066000 13.8000000

. 98786006.5999000 1.1106000 13.9000000

. 98722006.6999000 1.1140000 14.0000000

. 98665006.7999000 1.1170000

6.8999000 1.11950006.9999000 1.12160007.0999000 1.12330007.1999000 1.1246000

3 =0.6

- 195 -

t

secondsc(t)

t

secondsc(t)

.0000000 .0000057 7.0999000 1.0870000

. 1000000 . 0012293 7.1999000 1.0890000

. 2000000 .0048075 7.2999000 1.0907000

.3000000 .0105890 7.3999000 1.0921000

.4000000 .0184340 7.4999000 1.0932000

.5000000 .0282030 7.5999000 1.0940000

. 6000000 .0397640 7.6999000 1.0945000

. 7000000 .0529780 7.7999000 1.0947000

.8000000 .0677210 7.8999000 1.0948000

.9000000 .0838670 7.9999000 1.0945000

1.0000000 .1013000 8.0999000 1.0941000

1.1000000 .1198900 8.1999000 1.0935000

1.2000000 .1395400 8.2999000 1.0926000

1.3000000 .1601200 8.3999000 1.0916000

1.4000000 . 1815500 8.4999000 1.0905000

1.5000000 .2037200 8.5999000 1.0891000

1.6000000 .2265300 8.7000000 1.0877000

1.7000000 .2498800 8.8000000 1.0861000

1.8000000 . 2736900 8.9000000 1.0844000

1.9000000 .2978800 9.0000000 1.0826000

2.0000000 .3223700 9.1000000 1.08070002.1000000 .3470700 9.2000000 1.0787000

2. 2000000 .3719200 9.3000000 1.0766000

2.3000000 .3968400 9.4000000 1.0745000

2.4000000 .4217800 9.5000000 1.0723000

2.5000000 .4466600 9.6000000 1.07010002.6000000 .4714400 9. 7000000 1.06780002.7000000 .4960600 9.8000000 1.0655000

2.8000000 .5204700 9.9000000 1.06320002.9000000 .5446200 10.0000000 1.06080003.0000000

. 5684700 10.1000000 1.0584000

3.1000000 .5919700 10.2000000 1.05610003.2000000

. 6151000 10.3000000 1.05370003.3000000 .6378100 10.4000000 1.05140003.4000000 .6600800 10.5000000 1.04900003.5000000 .6818800 10.6000000 1.0467000

3.6000000 .7031800 10.7000000 1.0444000

3.7000000 .7239600 10.8000000 1.0421000

3.8000000 .7442000 10.9000000 1.0399000

3.9000000 .7638800 11.0000000 1.03770004.0000000 .7829900 11.1000000 1.03550004.1000000 .8015100 11.2000000 1.03330004.2000000 .8194300 11.3000000 1.03130004.3000000 .8367500 11.4000000 1.02920004.4000000 .8534600 11.5000000 1.02720004.5000000 .8695500 11.6000000 1.02530004.6000000 .8850200 11.7000000 1.02340004.7000000 .8998600 11.8000000 1.02150004.8000000 .9140900 11.9000000 1.01980004.9000000 .9276900 12.0000000 1.01800005.0000000 .9406800 12.1000000 1.01630005.1000000 .9530500 12.2000000 1.01470005.2000000 .9648200 12.3000000 1.01320005.3000000 .9759900 12.4000000 1.01170005.4000000 .9865700 12.5000000 1.01020005.5000000 .9965600 12.6000000 1.00890005.6000000 1.0060000 12.7000000 1.00750005.7000000 1.0148000 12.8000000 1.00630005.8000000 1.0231000 12.9000000 1.00510005.8999000 1.0309000 13.0000000 1.00390005.9999000 1.0381000 13.1000000 1.00280006.0999000 1.0449000 13.2000000 1.00180006.1999000 1.0511000 13.3000000 1.00080006.2999000 1.0568000 13.4000000 .99991006.3999000 1.0621000 13.5000000 .99904006.4999000 1.0669000 13.6000000 .99822006.5999000 1.0713000 13.7000000 .99746006.6999000 1.0752000 13.8000000 .99675006.7999000 1.07870006.8999000 1.0819000

6.9999000 1.0846000

- 196 -

A 13

t ct(t) t ct(t)seconds $ = 0.2 seconds \ = 0.2

.0000000 -

. 0000045 7.2000000 - .0934360

.2000000 . 0489270 7.4000000 - .1132100

.4000000 .0954610 7.6000000 - .1311100

.6000000 .1392400 7.8000000 - .1470200

.8000000 .1799300 8.0000000 - .1608600

1.0000000 .2172700 8.1999000 - .1725900

1.2000000 .2510100 8.3999000 - .1821700

1.4000000 .2809700 8.5999000 - .1895800

1.6000000 .3070100 8.7999000 - .1948500

1.8000000 .3290200 8.9999000 - .1980100

2.0000000 . 3469400 9.1999000 - .1991000

2.2000000 .3607600 9.3999000 - .1981900

2.4000000 .3705100 9.5999000 - .1953900

2.6000000 .3762400 9.7999000 - .1907800

2.8000000 .3780700 9.9999000 - .1844800

3.0000000 .3761200 10.2000000 - .1766200

3. 2000000 .3705600 10.4000000 - .1673400

3.4000000 .3615900 10.6000000 - .1567800

3.6000000 .3494300 10.8000000 - .1451100

3.8000000 .3343300 11.0000000 - .1324700

4.0000000 .3165400 11.2000000 - .1190200

4.2000000 .2963500 11.4000000 - .1049400

4.4000000 .2740500 11.6000000 - .0903860

4. 6000000 .2499400 11.8000000 - .0755150

4.8000000 .2243300 12.0000000 - .0604880

5.0000000 .1975300 12.2000000 -. 0454560

5.2000000 .1698400 12.4000000 - .0305700

5.4000000 .1415700 12.6000000 - .0159680

5.6000000 .1130200 12.8000000 - .0017816

5.8000000 .0844920 13. 0000000 .0118660

6.0000000 .0562510 13.2000000 .0248620

6.2000000 .0285650 13.4000000 .0371050

6.4000000 .0016866 13.6000000 .0485040

6. 6000000 .0241550

6.8000000 .0487470

7.0000000 .0718970

197 -

A 14

"(bo .2000000 E.g.3300000

2.0000000 .3469400 2.0000000 .3030300

2.1000000 .3543600 2.1000000 ,3076900

2.2000000 .3607(00 2.2OOOOO0 .311*500

2.3000000 .3661*00 2.30OO0O0 .31*3000

2.4000000 .3705100 2.4000000 .3162900

2.5000000 .3738700 2.5000000 .3174300

2.6000000 .3762400 2.6000000 .3177600

2.7000000 .3776300 2.7OOO0O0 .3172900

2.8000000 .3780700 2.8000000 .3160600

2.5000000 .3775500 2.9000000 .3140900

3.0000000 .3761200 3.0000000 .3114200

0.2500000 E- 6.4000000

2.0000000 .3313400 2.0000000 .2901700

2.1000000 .3377400 2.1000000 .2941200

2.2OOOOO0 .3431400 2.2OOOO0O .2971900

2.3O0O0O0 .3475600 2.3OOOO0O .2994100

2.4000000 .3510200 2.4000000 .3008200

2.5000000 .3533200 2.5000000 .3014300

2.6000000 .3550900 2.6000000 .3012800

2.7000000 .3557400 2.70O0OO0 .3004000

2.8000000 .3555000 2.80OOOO0 .2988100

2.9000000 .3543900 2.9000000 .2965600

3.0000000 .3524300 3.0000000 .2936700

E_B.3000000

2.0000000

2.1000000

2.2000000

2.3000000

2.4000000

2.5O0OOO02.6OOO0O0

2.7OOO0OO

2.8000000

2.9000000

3.0000000

{ =0-5000000

.3167300

.3222100

.3267400

.3303200

.3329900

.3347600

.3356500

.3356900

.3349000

.3333100

.3309400

I 0.4500000

2.0000000 .2780900

2.1000000 ,2814100

2.2000000 .2838800

2.3000000 .2855600

2.4000000 .2864600

2.5OO0O0O .2866300

2.6000000 .2860800

2.7000000 .2848700

2.8OO0O0O .2830000

2.9OOOOO0 .2805300

3.0000000 .2774800

f = 0.6000000

2.0000000

2.1000000

2.20OOOO0

2.3O0O0O0

2.4000000

2.5000000

2.6000000

2.70000002.8000000

2.9000000

3.0000000

.2667300

.2695000

.2714600

.2726600

.2731300

.2729200

.2720500

.2705600

.2684900

.2658600

.2627100

2.0000000 .2460600

2.1000000 .2478700

2.20OOOOO .2489700

2.3OOOO0O .2494100

2.4O0OOOO .2492100

2.5O0O0OO .2484300

2.6000000 .2470800

2.7000000 .24522002.80O00OO .24286002.9O0OOO0 .2400600

3.0000000 .2368400

| iO .5500000 f : O.650OO0O

2.0000000

2.10000002.2000000

2.3000000

2.4000000

2.5000000

2.6O0OO0O

2.70OOOO0

2.8O0OO0O

2.9000000

3.0000000

.2560900

.2583400

.2598400

.2606300

.2607400

.2602100

.2590800

.2573800

.2551500

.2524100

.2492100

2.0000000

2.1000000

2.2000000

2.3000000

2.4000000

2.5000000

2.6000000

2.70OOOO0

2.8000000

2.9000000

3.0000000

.23(6100

.2380400

.2388000

.2389300

.2384700

.2374700

.2339600

.2339700

.2315400

.2287000

.2254900

- 198 -

A 15

X *1seconds seconds

*3seconds

T— sees.

2

.1000000 3.3588000 9.6736000 15.9890000 6.3154

. 1250000 3.4191000 9.7520000 16.0850000 6.3330

. 1500000 3.4821000 9.8373000 16.1920000 6.3547

. 1750000 3.5482000 9.9299000 16.3120000 6.3821

.2000000 3.6174000 10.0300000 16.4430000 6.4130

. 2250000 3.6901000 10.1390000 16.5870000 6.4480

.2500000 3.7666000 10.2560000 16.7450000 6.4890

.2750000 3.8471000 10.3820000 16.9170000 6.5350

.3000000 3. 9321000 10.5190000 17.1050000 6.5860

.3250000 4.0220000 10.6660000 17.3100000 6.6440

.3500000 4.1172000 10.8250000 17. 5320000 6.7070

.3750000 4.2182000 10.9960000 17.7740000 6.7780

.4000000 4.3258000 11.1810000 18.0370000 6.8560

.4250000 4.4405000 11.3820000 18.3230000 6.9410

.4500000 4.5633000 11.5990000 18.6350000 7.0360

.4750000 4.6950000 11.8350000 18.9750000 7.1400

.5000000 4.8368000 12.0920000 19.3470000 7.2550

.5250000 4. 9901000 12.3730000 19. 7550000 7.3820

. 5500000 5.1563000 12.6800000 20.2030000 7.5230

. 5750000 5.3374000 13.0170000 20. 6970000 7.6800

. 6000000 5.5358000 13.3900000 21.2440000 7.8540

- 199 -

Appendix B

Refer to section (7.2.0).

Given:-

Cj(t) = ^e"*' sin pt (b.l)

c(t) = X +-^_ e"0Ctsin(pt + >V) (b.2)

fl*

(It - tan"1 £)X(oc>P) =

»* \ -,-S (b.3)

(*" + pi sin tan1

£-

cd(t) = cU+tj) - -°^ (b.4)

Po

Prove: -

cd(t) = X(«, p) • c.(t) (b.5)

Proof:-

Form equations (b. 1) and (b. 3)

x««.„ •

ci(„.

ytg-^t' s.-«mpi1 (ot2 + p"2) sin tan

X -£- P

* (b.6)

sin tan-1 -§- =,L. and (oc2 + p 2//2 = pn (b.7)

*(oc2+p2)/2

°

Substituting equation (b. 7) into equation (b. 6) gives

o -5-(1l - tan"1-&-) a

X(cc.p) • ci(t) = P,e Y 2

* ^ e-atsin pt

(062+pa)p(«:J+P2)-,/2P

(b8)

- 200 -

ae-|(H- tan"1 A)X(ot, (i) • c(t) =

-2 e"wtsin fit (b.9)

PPo

From equation (b. 4) and from for t- = -

-jr, c .(t) may be written as

cd« = \ + i" e-^-i)sin[P(t-^)+Y]- \(*o ' ' o |i0

= A-e^e-f ("^sinpt (b. 10)

Comparing equations (b.9) and (b. 10), it is seen that if IT - tan- -L-= --vy, then

the two equations are identical. From equation (7.1.8) and since a1 = 0, therefore

t = - tan-1 JL (b. 11)

-y = tan"1 ^ (b.12)

But

TU - tan"1 £•= tan"1 JL (b.13)

therefore,

cd(t) = X(o6,p) • Cj(t) (b.14)

Q.E.D.

- 201 -

ZUSAMMENFASSUNG

Gibt man auf ein lineares System die Eingangssignale r^t).. .rk(t), die der Be-

dingung

T T T

r n fn r n

J ri(t)dt = J r2(t)dt =... = J rfc(t)dt

Tn-1 Tn-1 Tn-1

geniigen, wobei (T -T .) Zeitintervalle sind, welche kleiner sind als die Halfte der

kleinsten dominierenden Zeitkonstanten des Eingangssignals und des Systems, so wer-

den die Ausgangssignale annahernd die gleiche Form haben. n ist eine ganze Zahl,

welche alle Werte annehmen kann, um die in Frage stehende Zeitdauer zu iiberstrei-

chen. Die Gute der AnnSherung hSngt von der Intervalldauer (T -T *) ab, je kleiner

diese gewalilt wird, umso hoher ist die Genauigkeit. Um untersuchen zu konnen, wel¬

che Auswirkungen die Modilikation des Eingangssignals eines linearen Systems auf

den Ausgang hat, wurden zwei transistorisierte Signalgeneratoren konstruiert. Dire

Hauptvorziige liegen darin, dass sie eine ausserst grosse Zahl von verschiedenen Test-

signalen erzeugen konnen und trotzdem ein kleines Volumen aufweisen.

Das Uebergangsverhalten eines Regelsystemes kann dadurch verbessert werden,

dass es vermittels eines Umschalters abwechslungsweise iiber- oder unterkritisch ge-

dampft arbeitet. Dadurch werden die Anstiegszeit und ebenso die gesamte Regelzeit

bis zum Erreichen des Endwertes herabgesetzt. Im Falle von getasteten Rejelsyste-

men kann eine Verbesserung der Uebergangsfunktion erreicht werden, wenn man Ta¬

ster mit zwei verschiedenen Tastdauern verwendet. Dies wird analytisch und mittels

Simulation auf einem Analogrechner gezeigt.

Die Uebergangsfunktion von Systemen zweiter Ordnung kann ferner dadurch ver¬

bessert werden, indem die Dampfung in Funktion der Eingangs- und Ausgangssignale

variiert wird. Bei getasteten Systemen erreicht man dasselbe durch Modulation von

Tastdauer und Tastfrequenz als Funktion der Ein- und Ausgangssignale.

Ferner wird gezeigt, dass die Uebergangsfunktion linearer Systeme verbessert

werden kann durch Verwendung von stufenfSrmigen Eingangssignalen. Es wurden

Systeme erster bis fiinfter Ordnung untersucht und in den meisten Fallen ist es mog-

lich, die Grfisse und Dauer der einzelnen Stufen mit Hilfe der beschriebenen graphi-

schen und analytischen Methoden zu bestimmen. Von einem zweistufigen Eingangssignal

- 202 -

werden die entsprechenden Pole und NuUstellen bestimmt. Fallen die NuUstellen des

Eingangssignals mit den Polen des Systems zusammen, so entsteht ein "deadbeat"

Ausgangssignal.

Dann wird der Begriff des allgemeinen Tasters eingefiihrt und es wird gezeigt,

dass durch Verwendung eines solchen als Pulsbreitenmodulator eine Verbesserung des

Ausgangssignals erreicht werden kann.

Schliesslich wird ein System zweiter Ordnung untersucht, dessen DSmpfungs-

koeffizient sinusformig oder stufenweise variieren kann und dessen Eingangssignal

aus zwei Stufen besteht. Es wird gezeigt, dass das Ausgangssignal befriedigend ist,

wenn man einen adaptiven Regler verwendet, der die relative GrSsse und Dauer der

Eingangsstufen einstellt. In seiner einfachsten Form besteht der adaptive Regler aus

einem Integrator.

- 203 -

Bibliography

(1) R. Andeen, Doctoral Dissertation, Northwestern University, Illinois, 1958.

(2) R. Andeen, "Analysis of Sampled Data Systems with Nonlinear Elements",

Proc. of the IFAC Congress Moscow 1960, pp. 314-320.

(3) N.Billawala, "An Experimental Study of pulse-width modulated sampled

feedback control systems", University of California, M. S. The¬

sis 1958.

(4) R. R. Caldweel, V. C. Rideout, "A Differential Analyzer Study of Cer¬

tain Nonlinearly Damped Servomechanisms", Trans. AIEE, Pt.

n, July 1953.

(5) S. S. L. Chang, "Synthesis of Optimum Control Systems", McGraw Hill

Book Co., Lie. New York, 1958.

(6) B. Chatterjee, "Improved Transient Response in Servo Systems with In¬

put Modification", IRE Trans, on Automatic Contro}, June

1960, pp. 152-153.

(7) H. Chestnut and R.Mayer, "Servomechanisms and Regulating System

Design", Vol. I, John Wiley and Sons, Inc. New York 1959.

(8) F.R.Delfeld and G.J. Murphy", Analysis of Pulse-Width Modulated

Control Systems ", IRE Trans, on Aut. Control, vol. AC-6,

Sep. 1961, pp. 283-292.

(9) D. J. Gimpel, J. F. Calvert, "Signal Component Control", Trans AIEE,

vol. 71, pt. n, Nov. 1952, pp. 339-343.

(10) G. C. Gupta and E. I. Ju ry, "Statistical Study of Pulse-Width Modulated

Control Systems", Journal of the Franklin Institute, April 1962,

pp. 292-321.

(11) M. Hamza, "Compensation using pulse-width modulation in automatic control

systems", New Techniques, vol. 6, 1961, pp. 397, 398. A

short note.

(12) M. Hamza, "Adaptive System Controllers using Delay Elements", Proc. of

the First International IFAC Symposium on Optimizing and

Adaptive Control held in Rome April 1962. Published by ISA,

April 1963.

- 204 -

(13) M. Hamza, "Sampling as a Means of Compensating Control Systems,", Proc.

of the First IFAC Symposium on Relay Systems and Finite Auto¬

mata Theory, Moscow Sep. 1962. To be published by the Acad,

of Sciences USSR and by ISA.

(14) M. Hamza, "The Chronological Development of Optimum and Self-Adjusting

Systems", presented at the Technical Congress on the occasion

of the 2 International Exhibition of Laboratory, Measurement

and Automation Techniques in Chemistry, held in Basel Oct.

1962. To be published in New Techniques, vol. 4, 1963.

(15) Prof. Hildebrand's course on "Methods of Applied Mathematics", offered

atM.I.T. in 1957.

(16) E. Holzler, H. Holzwarth, "Theorie und Technik der Pulsmodulation",

Springer-Verlag, Heidelberg 1957.

(17) E.I. Jury, "Sampled Data Control Systems", John Wiley and Sons, Inc.,

New York 1958.

(18) T.Kadota, Doctoral Dissertation, University of California, Berkeley 1959.

(19) T. Kadota, H. C. Bourne Jr., "Stability Conditions of Pulse-Width Mo¬

dulated Systems through the second Method of Lyapunov", IRE

Trans, on Aut. Control, vol. AC-6, Sep. 1961, pp. 266-276.

(20) G. G. Lendaris, O. J. M. Smith, "Complex-Zero Signal Generator for

Rapid System Testing", Trans. AIEE, Vol. 77, pt. H, pp. 534-

540.

(21) J. B . Lewis, "The Use of Nonlinear Feedback to Improve the Transient Res¬

ponse of a Servomechanism", Trans. AIEE, pt. n, Jan. 1953.

(22) J.M.Millman, H.Taub, "Pulse and Digital Circuits", McGraw Hill

Book Co., Inc. New York 1956.

,(23) E

. Mishkin, L. Braun Jr., "Adaptive Control Systems", Mc Graw Hill

Book Co., Inc. New York 1961.

(24) R.Nease, "Analysis and Design of Nonlinear Sampled-Data Control Systems?,

Technical Note 57-162. Servomechanism Laboratory M. I. T.

(25) W.Nelson, "Doctoral Dissertation, Columbia University, New York 1959.

(26) A. I. Pressman, "Design of Transistor Circuits for Digital Computers",

John F. Rider Publisher, Inc. New York 1960.

(27) B.N.Petrov, "The Invariance Principle and the Conditions for its Applica¬

tion during Calculation of Linear and Nonlinear Systems", Proc.

of the IFAC Congress Moscow 1960, pp. 117-126.

- 205 -

(28) J. R. Ragazzini, G. P. Franklin, "Sampled-Data Control Systems",

McGraw Hill Book Co., Inc. New York 1958.

(29) A. E. Rogers, T. W. Connolly, "Analog Simulation in Engineering De¬

sign", McGraw Hill Book Co., Inc. New York 1960.

(30) C. Y. She, "Analog Study of Posicast Control by Relay Systems", Proc.

IRE, Sep. 1961, pp.1450, 1451.

(31) O.J.M.Smith, "Feedback Control Systems", McGraw Hill Book Co.,

Inc. New York 1958.

(32) H.C.So, G.J.Thaler, "A Modified Method of Control with Applications

to Higher-Order Systems,", Trans. AIEE, Applications and

Industry, Nov. 1960, pp. 320-326.

(33) G. H. Tallman, O. J. M. Smith, "Analog Study of Deadbeat Posicast

Control", IRE Trans, on Automatic Control, March 1958,

pp.14-21.

(34) G. J. Thaler, M. P. Pastel, "Analysis and Design of Nonlinear Feedback

Control Systems", McGraw Hill Book Co., Inc. New York 1962.

(35) J. T. Tou, "Digital and Sampled-data Control Systems", McGraw Hill Book

Co., Inc. New York 1959.

(36) J.G.Truxal, "Automatic Feedback Control System Synthesis", McGraw

Hill Book Co., Sic. New York 1955.

(37) J. Zaborsky, "A simple Nonlinearized Control System", Trans. AIEE, vol.

77, pt. H, pp. 430-436.

Curriculum vitae

Born in Heliopolis Egypt on October 17,1936. I attended "Woodley School

Heliopolls" from 1942-1946, "Victoria College Cairo" from 1946-1950, and "The

English School Heliopolis" from 1950-1955. I passed the London University General

Certificate of Education examination, ordinary level, in 1953 and advanced level in

1954. In 1955 I passed the Oxford University General Certificate of Education exa¬

mination advanced and scholarship levels.

In September 1955 I started my studies at "The Massachusetts Institute of

Technology". I was awarded the S. B. degree in Electrical Engineering in February

1958. Following graduation, I spent one more semester at M. I. T. attending advan¬

ced courses. In July 1958 I returned back to Egypt.

During January and February 1959 I was at the "Goethe Institute" in Kochel,

Germany. In April 1959, I began my studies at "The Swiss Federal Institute of Tech¬

nology". I joined the Institute for General Electrical Engineering - later called the

Institute for Automatic Control and Industrial Electronics - under the guidance of

Prof. Ed. Gerecke. In April 1960 I passed the admission to the doctorate examina¬

tion. Since that time up to the present, February 1963, I performed research work

in the fields of sampled-data, nonlinear, optimum and adaptive control systems.

Besides this dissertation, I have a paper which is published as part of the Procee¬

dings of the International Symposium on Optimizing and Adaptive Control held in

Rome, April 1962. I presented a paper at the International Symposium on Relay

Systems and Finite Automata Theory held in Moscow during September 1962; fur¬

ther, I presented a paper at the technical congress on the occasion of the 2 Inter¬

national Exhibition of Laboratory, Measurement and Automation Techniques in Che¬

mistry which was held in Basel in October 1962.