ON FRG?ER?IES OF A CLASS OF SYS'E'EMS 0F
ETIEFFEREKTEAL EQUATE’QNS ANS CQRRESPONDING
mask-Gama DEFFERENTIAL EQUATIONS

find: for the Degree of Ph. D.
MICHIGAN STATE UNIVERSN'Y

Roberf Hampfen Ragers
i963

THESIS

   
 

  

LIBRARY

Michigan State
- University

 
   

 

This is to certify that the

thesis entitled

ON PROPERTIES OF A CLASS OF SYSTEMS OF
DIFFERENTIAL EQUATIONS AND CORRESPONDING
HIGHER-ORDER DIFFERENTIAL EQUATIONS

presented by

Robert Hamptom Rogers

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Elect. Engrg.

I
Major professo’FP 1

Date Anggst 8. 1963

0-169 ¢

.5__ ._ » ».—. , - __ , 7 i ,_i ,. ---‘-'—rrwr—--—-o __.'_V _r~+f --.—.-._-_,V -7 V ,,, i

ABSTRACT

ON PROPERTIES OF A CLASS OF SYSTEMS OF
DIFFERENTIAL EQUATIONS AND CORRESPONDING HIGHER—
ORDER DIFFERENTIAL EQUATIONS

by Robert Hampton Roge rs

Procedures have recently been presented for formulating
time-(loniain models of linear and nonlinear systems in the form

(1

El—t—Xi '— {i (X

1,... 1’1

This set of n differential equations is referred to as the normal

system or state model. Procedures for determining the solution

of the system and hence analyzing the. system performance presently
employ analog computers, digital computers, and/or functions of
Inatrices. The Choice, of mathematical procedures to apply in the
design of a particular physical system varies from problem to prob-
lem. However, in all Cases the objective. is to gain information or
knowledge pertaining, to the inter-relationship of the system parameters
to the system performance.

A knowledge of this inter-relationship is obtained in the
thesis by formally developing the. mathematical properties which
relate the parameters in the normal system of linear and a class of
nonlinear differential equations to the parameters in an r-order

(r<_ n) differential equation.

Abstract

(‘v

Robert H. Rogers

The mathematical foundation of the thesis is established
by developing the mathematical properties which relate the solution
of the normal system to the solution of an r-order (r E n) differential
equation obtained from the system. This development is based on
deriving the r-order equation from the normal system by means
of a certain nonsingular transformation. In this development,
conditions on the parameters of the normal system are determined
so that an n-order differential equation is obtained from the system.
In the proof of these results a technique for formulating a nonsingular
transformation is given which allows the determination of the solution
of the normal system in terms of the solution of an n-order dif-
ferential equation.

The mathematical properties developed in the thesis
are applied in the formulation of methods for the design of physical
systems. The design methods necessitate constructing : (l) a
function y(t) from the specification of a desired system performance
and (2) a normal system of differential equations having the function
as a component of the system solution.

Two methods of constructing the linear system, to
have a specified solution, are given. One method consists of
determining the coefficients and initial conditions of the normal
system in terms of the coefficients and initial conditions of an n-
order differential equation. A second method relates the coefficient
matrix in the normal system directly to the specified solution y(t)
by means of a certain matrix transformation. If the normal system
is nonhomogeneous then an explicit formula is given for determining

the nonhomogeneous part of the system in terms of the specified

Abstract 3 Robert H. Rogers

solution y(t). Similar results are developed for constructing a
special class of nonlinear differential equations having a specified
solution.

The design methods, proposed in the thesis, are

illustrated in the design of amplifiers and oscillators in the time-

domain.

ON PROPERTIES OF A CLASS OF SYSTEMS OF
DIFFERENTIAL EQUATIONS AND CORRESPONDING

HIGHER-ORDER DIFFERENTIAL EQUATIONS

By

Robert Hampton Rogers

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Electrical Engineering

1963

Few-nae

7/8/69 4

AC KNOWLE DGEM EN T

The author wishes to express his appreciation to
Dr. Dave P. Brown for his advice and guidance in preparing this
thesis. He also wishes to thank Dr. William A. Blackwell for his

patience in its editing.

Finally, the writer is indebted to his wife, Sandra
and family for their understanding, patience and moral support

during graduate study .

ii

II.

III.

IV.

VI.

TABLE OF CONTENTS

INTRODUCTION

PROPERTIES OF SYSTEMS OF HOMOGENEOUS
DIFFERENTIAL EQUATIONS AND ASSOCIATED

HIGHER —ORDER DIFFERENTIAL EQUATIONS ,
PROPERTIES OF SYSTEMS OF NONHOMOGENEOUS
DIFFERENTIAL EQUATIONS AND ASSOCIATED HIGHER
ORDER DIFFERENTIAL EQUATIONS

ON A CLASS OF SYSTEMS OF NONLINEAR DIFFERENTIAL
EQUATIONS AND CORRESPONDING REDUCED
DIFFERENTIAL EQUATIONS

ON DESIGN METHODS AND EXAMPLES

CONCLUSION

LIST OF REFERENCES

iii

Page

28

46

58
82

85

LIST OF APPENDICES

APPENDIX Page
A. Sonie Theorems and Definitions from References 86
Solutions of Certain Differential Equations . 89
C. On a Special Inter-relationship of a M—Order

Polynomial to a Sine Function , , , 100

iv

I. INTRODUCTION

 

The design of a physical system to meet a given
performance specification necessitates decisions based on a
knowledge of the inter-relationship of the system components
(parameters) to the system performance. A major portion of
any design procedure is devoted to the process of relating the
system parameters to the system performance.

Brown“), Wirth(2), Koenig, Tokad, and Kesavan(3)
have recently presented methods of formulating the mathematical

model of a physical system into the form

dx.__ . -_
WITfi<X1P ,,,xn,t),1_l,2,. . .,n. (1.1)

The set of differential equations (1. l) is commonly referred

to as the normal system, the linear form of this system will be

 

denoted by
X : AX + Q(t) (1.2)

where X' : [xl,x. , . . . ,xn] , A : [aij] a square matrix of order
n, Q'(t) : [ql(t), q2(t), . . . , qn(t)] and the prime denotes the
transpose.

One method of approaching the design problem, when
presented with a mathematical model of the system in the normal
form, is to view the system performance as one specified com-
ponent xi(t) of the vector solution. Then relate the parameters

1

in the normal system to the system performance xi(t). One technique
of obtaining these parameter-solution relationships (design equations)
is to (1) convert the normal system into a higher-order differential
equation in Xi by a reduction method such as the one proposed by
Murray and Miller [4, p. 126], and (2) determine the inter—relation-
ship of the system performance xi to the system parameters in the
higher-order differential equation. An obstacle to obtaining the
desired parameter-solution relationships by this technique is that
the reduction method may convert the normal system into a differen-
tial equation of r-order, where the number r is less than the number
of equations in the normal system, i.e. r _<_ n. Moulton [5, p. 9]
presents, as an example, a normal system of three equations which
converts into a second—order equation in any component, xi. In
seeking these parameter-solution relationships, many questions have
arisen concerning the mathematical properties which relate the
normal system (1.1) to an r-order (r E n) differential equation

1' dy dr-l

:f(y,——,.. .,—-———y; t) (1.3)
dtr dt dtr-1

 

obtained from the system.

The objective of this thesis is to develop the math-
ematical properties which relate the normal. system (1.1) to an
r-order (rin) differential equation (1. 3). The mathematical
properties, thus obtained, are to afford new tools for the design
of electrical networks.

When the mathematical model has the linear form (1. 2)
certain questions arise as to how the solution of the normal system

is related to the solution of the r-order (r_<_n) equation (1. 3) obtained

from the system.

Coddington and Levinson [6,p. 21] as well as others

[7, p. 33] have applied the nonsingular transformation

ixll i—y 1
(1)
x y
2 z . (1.4)
(“-1)
txnl -y -

 

 

 

 

to the normal system (1. 2), where matrix A has the special

form of a companion matrix (2. 2. 4) and Q'(t) : [0, O, . . . , O,qn(t)]
to obtain an n-order differential equation. The n-order equation
obtained from the system by the transformation (1.4) is referred
to by Coddington and Levinson as "associated" with the system.
The nonsingular transformation (1.4) thus specifies the manner in
which the solution of the normal system and the solution of the n-
order differential equation associated with the system are related.
A generalization of this (nicept of the associated differential
equation is required in Chapter II (homogeneous) and Chapter III
(nonhomogeneous) to develop the general mathematical properties
which link the solution of the normal system (1. 2) to the solution of
the r-order differential equation obtained from the system.

The later parts of Chapters 11 and III develop techniques
for relating the parameters in the system (1.2) to one specified
component xi(t) of the vector X(t) solution. The parameter-solution
relationships developed in these sections have not been formally
investigated and made available to the network designer until now.
These relationships, as shown in Chapter V, offer new tools for the

design of electrical networks.

In Chapter IV mathematical relationships are developed
which relate the solution of a class of nonlinear systems (1.1)
to a solution of an n-order differential equation (1. 3), obtained
from the system. A portion of the results obtained in Chapter IV
are applied to the design of tunnel-diode amplifiers and oscil-
lators in Chapter V. The design equations obtained in Chapter V
are shown to be a generalization of results obtained by Kim [8, p.416]

by a different Inethod.

PROPERTIES OF SYSTEMS OF HOMOGENEOUS

 

 

 

II.
DIFFERENTIAL EQUATIONS AND ASSOCIATED
HIGHER ORDER DIFFERENTIAL EQUATIONS
.1 Introduction

 

The general mathematical properties that relate the

solution of the normal system

 

 

 

 

 

j —' .. r-
(X1 “‘11 8112 . . . aln x1l
d X2 a21 a22 ' ' ° aZn X2
3 :
L xn J Lanl an2 annJ _ XnJ

 

where the entries are constants or in symbolic form
(2.1.1)*

XZAX

to the solution of the r-order (r<n) homogeneous differential

 

equation
r r r-i
(3—1; 2 1: ai d rd" (2.1.2)
(it izl dt

obtained from the system are developed as a result of deriving

the r-order equation from the system by means of the transformation

(2.1.3)

,x 1, Y' : [o,o,...,o,y,y(l,)...,yfr'Do,o,...01.

whereX : [xl,x2,... n

 

The number indicates the section and the number of the
Thus (2.1. 1) indicates Sec. 2.1,

:1:

equation in the section.
Eq. 1.
5

6

and the -1 superscript indicates the inverse of matrix C.

In the investigation of the general solution properties
a number of problems have arisen for which the solution will be

of interest to the system designer.

System design methods are proposed in Chapter V that
necessitate constructing: (1) a function y(t) from the specification
of a desired system performance and (2) a normal system (2.1.1)

having the function y(t) as a component of the system solution, X(t).

The construction of the normal system (2. 1.1) to have
the component y(t) as a solution necessitates a knowledge of how the
component y(t) is related to : (1) the solution of the normal system

X(t) and (2) the aij entries in matrix A in the normal system.

It is shown in the development of this chapter that these
parameter-solution relationships can be obtained by viewing the
function y(t) as the solution of the r-order differential equation

(2.1.2).

In Section 2. Z, the solution of the normal system is
shown to be related to the solution of the r-order differential
equation by the transformation (2. l. 3). The formal development
of this transformation allows the system designer to establish the
initial conditions on the normal system in terms of the specified
component y(t). In this development it is shown that y(t) is the

component of the system solution denoted by xn(t).

The problem of relating the ajj entries in the matrix

A in the normal system (2.1. l) to the solution y(t) is developed in

Section 2. 3. One method requires constructing the r-order dif-
ferential equation to have the specified solution y(t), and then re -
lating the coefficients aj j = 1, 2, . . . ,r of the r-order differential
equation to the entries a” in the normal system. A second method
relates the coefficient matrix A in the normal system (2.1.1)
directly to the specified solution y(t) by means of a certain matrix
transformation.

The parameter-solution relationships formally developed
in this section are referred to as design equations when applied to

system design in Chap. V.

2. 2 Systems of First Order Homogeneous Differential Equations

 

and Associated Higher Order Differential Equations

 

In this section it is shown that when the r-order equation
(2.1. 2) is obtained from the normal system (2. 1. 1) by a nonsingular
transformation of the form (2.1. 3), that the solution of the system and
the solution of the r-order equation are linked by two well-defined

properties.
Property (a) The solution of the r-order equation is

given by r entries in some row of matrix triple product
CFmD when C, F and D are nonsingular and each column
m
of F is a solution of the normal system.
m
. . -1
Property (b) A solution of the system is given by C Y
where Y is a VECtOI‘ containing the solution and derivatives

of the solution of the r-order differential equation.

8
The r-order equation defined by Properties (a) and (b) is said to
be "associated" with the system. These properties are stated

mathematically by the. following definition;

Definition 2. 2. 1: Consider the homogeneous system of differential

 

equations :

)< 2: 1A}( (2.22.1)

where A : [a..] , X' : [xl,x

1] 2,...,x].

II

An r—order homogeneous differential equation

 

 

(2. 2. 2)

I

is associated with (2. 2.1) if,

(a) for Fm a fundamental matrix* of (2. 2. 1), there exists
non-singular matrices C and D such that the (i,i ), (i, i+l), . . . ,
(i, i+r-—1) entries in CEnD, for some i, i: 1, Z, . . . , n, are a

fundamental set of (2.2. 2) and

(b) for yl, yz, . . . , yr a fundamental set of (2. 2. 2), the
-1 . (r-D
vectors C Yj where Yj : [O’O’°”'O’-\:j'yj"”’ yj 0,0,...,0],

j z 1, Z, . . . , r, are r linearly independent solution on the open

interval I defined by I : [tzt < t < t2], t and t are constants, of

1 1 2

(2.;a.1).

 

* For the definitions of terminology and notations used
throughout, see Appendix A.

The mathematical properties specified in Def. 2. 2.1
are first encountered in Thm. 2. 2.1 when a transformation of the
form (2.1.3) is applied to a normal system (2. 2.1) to obtain a
set of s differential equations, lisinmf the form (2. 2. 2). This
set of s, higher-order differential equations, are ascertained to

be associated with the system.

Before proving the existence of a set of 5 higher—
I
order differential equations of the form (2. 2. 2) associated with
the system (2.2.1), some basic relationships must be established
between a fundamental matrix of the system (2. 2.1) and a funda-

mental matrix of the transformed system (2. 2. 3). Lemma 2. 2.1

follows immediately from results established in Thm. 2. 2 [6, p. 69].

Lemma 2. 2.1:. If Fm is a fundamental matrix of the

 

system (2.2. l) and Y : CX, where C.1 exists, then CFm is a

fundamental matrix of

Y = CAC’IY (2.2.3)

Proof: Let fj, j : 1,2, . . . ,n be the j-column of Fm
Since fj is a solution on I of (2. 2.1) and (2. 2. 3) is obtained by sub-
stituting X : C_1Y in (2. 2.1), ij is a solution of (2.2.3) on I.

Since C and Fm are nonsingular by hypothesis, CFm is nonsingular.

The lemma follows .

10

Lemma 2. 2. 2: Suppose CAC-l of Lemma 2. 2.1 is

 

 

 

,
[B1 0 ... o
0 2B . o
CAC—l: Z
0 o B
L a

where Bi is of order ri, i : 1, 2, . . . , 8. There exists a nonsingular

matrix D such that a fundamental matrix of Y1 : BiYi’ i : 1, 2, ..., s

where Y' : [Yl', Y', . . . . Y' ]. is the submatrix of CFmD consisting

2 . s .
1—1 1-1 1
of the entries in rows er+1, :3 r‘j + 2, . . . , 23 r. and columns
i-l 1—1 3:1 1 3:1 3:1
Er.+1, Er.+2,...,2 r..
j:1 3 j=1 3 J31 J
; Proof: If Ci is a fundamental matrix of Yi :: BiYi’ 1?..1, 2, . . . ,s,

then a fundamental matrix of (2.2. 3) is

 

 

c ... 0
e1 0
o o . o
_ 2
Gig. ..
o o .. . o,
h S-d

By Lemma 2. 2.1 CFm is a fundamental matrix of (2. 2. 3).

Therefore, by Thm. A. 1, there exists a nonsingular matrix D

such that
G : CF D
ls m
i-l i-1

Since the entries in Gi are the entries in rows )3 r.+l, 2: r.+2, . . . ,
i 31-1 1-1 1 j:1 J jzl J

E r. and columns 23 r.+ l, E .r.+2.,...,3 r. of CF D, the lemma

.2 j ._1j ,m J g" m
J l J- J~l J~1

follows .

11

Theorem 2. 2. 1: There exists a set of s, homogeneous

 

differential equations, l_<_ s <n of order ri, i = l, 2, . . . , s,
5
associated with the system (2. 2. 1) such that E ri : n.
i=1

Proof: Substitute X : C—lY in (2. 2. 1) so that (2. 2.3)

results where

 

 

B1 0 o)
CAC—l: 0 B2 . 0
L0 0 B-
4 b‘
and
[o 1 0 o ‘
O 0 l 0
Bi:
0 0 O 1
a'r.r. ar.r.-—1 ar.r.-2 ' ° ' ar.1
L 1 1 1 l 1 l l .4

 

 

It has been shown [9, p.49] that such a transformation exists.

For any subset of equations of (2. 2. 3) of the form

Y. : B.Y. (2.2.4)
1 1 l

wh r .' z . . . . . . . . .
e 6 Y1 [yil’y12’ ’yiri], calculate ri—l successn'e derivatives

of the first row of (2. 2. 4) eliminating each time from the right hand

side the first derivatives of in’ y by means of the last
i

ri-l equations of (2.2.4). This process results in

i3’ yir

 

 

 

 

Substituting (2. 2. 5) into (2. 2.4) gives,

 

 

 

['(l) 1 F
Yil 0 0 O
(2)
yil 0 0 l .
(r-D
yill O 0 0
(ri)
.Yll ar.r. ar.r.-l a'r.r.-2
: ..1 L 1 1 1 1 1 1
The last row of (2.2. 6)
r1 r
d yil .}
r : 2" arj
(it i jzl i

1 - 1
[Yil y11
(l)
yi1 in
(ri-l)
[Yil ’ yiri

 

 

 

 

(2.2.5)
1 r 7
0 yil
(1)
0 yil
(2)
V11 (2.2.6)
1
(rf'l)
aril yil
1 L 1
r.-j
d 1 y11
r.-j (2.2.7)
dt 1

is associated with (2. 2.1). For if Fm is a fundamental matrix of

(2. 2.1), by Lemmas 2. 2.1 and 2. 2.2 andthe initial part of the

proof, there exists nonsingular matrices C and D such that a sub-

matrix of CFmD, Gi’ (Lemma 2. 2. 2)is a fundamental matrix of

(2. 2.4). By (2.2. 5) and (2.2. 6) each entry in row one of Gi is a

1 solution on I of (2. 2. 7) and the j-row of Gi is the j-l derivative of

the first row forj : 1,2,... ,ri.

From Thm. A.2, since the

l3
determinant of Gi is the Wronskian of solutions on I of (2. 2. 7)
and since Gi is nonsingular, the entries in row one of Qi are a

fundamental set of (2. 2. 7).

If yl,y2, . . . ,yr is a fundamental set of (2.2. 7), by

i
(2. 2. 6) and (2. 2. 5), ri linearly independent solutions on I of (2. 2. 4)
(r.-l)
are [yj,y;l), , yj1 ]' , j: 1,2,... , ri. This implies that the
1) ff” .
vectors Y3: 0,...,0, .,(. ,...,. ,0,....0, 21,2,...,r.
J [ YJ YJ YJ . 1 J 1

. . . . -l
are ri linearly independent solutions on I of (2. 2. 3). Since X :C Y,

ri linearly independent solutions on I, of (2. 2.1), are C-R’j, j: 1, 2,...,ri.
Therefore, (2. 2. 7) is associated with (2. 2.1). Corresponding to

each matrix Bi’ of order ri, i 2 1,2, . . . ,3, there exists an associated
differential equation of order ri. Since matrix A is of order n, then

s

E ri : n.
i:1
Theorem 2.2.1 establishes that, (2. 2.1), an arbitrary
system of n differential equations explicit in the first derivative ,
may be converted into a set of s, lfsfin, homogeneous differential
5
equations of order ri, i: 1,2, . . . ,s, where E ri : n (ri may be
i:l
less than n). This is a generalization of results found by Murray

and Miller [4, p.129].

In Theorem 2. 2. 2 which follows , the matrix product
CAC"1 is assumed to have the form of a set of 5,1:S_<_n: companion
matrices Bi’ i : 1,2, . . . ,s as in the proof of Thm. 2. 2.1. It is
then proved that the zeros of the characteristic equation of (2.2. 2)
an r-order differential equation, associated with a system of n
homogeneous differential equations, are also zeros of the character-

istic equation of (2. 2. l) the system of differential equations. These

14
properties are obtained by applying well-known results from

matrix algebra.

Theorem 2. 2. 2: Suppose

 

 

 

r r r-j
dr X = 3 d, x (2.2.8)
dt j.—.1 3 dt 3
is associated with

as in the proof of Thm. 2.2. 1. Then if 1. o is a zero of L0.) :
r

(-l)r [Kr - Z a). r-J] then 1.0 is also a zero of det [A-X I] :
i=1 3
Pr___o__of: By the method of the proof of Thm.2. 2.1
det [CACl — XI]=:[1| det i — RI]. By Lemma [6, p.88], the
r. r. r. r.-j
characteristic polynolmlial for Bi is det [Bi- XI] = (-l) 1D. 1- 211an 1 ],
1:]- _1

Since C is nonsingular, the zeros of det [A - XI] = 0 and det [CAC -)\I] : 0
are the same. .Therefore if x0 is a zero of L( ) it also must be a

zero of det [A - kl] : 0.

Corollary 2. 2. 1: With the same hypothesis and r : n,

 

a}. is (-l)‘]+1 times the sum of the principal minors of orderj of A.

Proof: This follows from Thni. 2. 2.2, Thm. A. 3 and

 

the fact that the coefficient of X n in det [A - XI] is (—l)n

Corollary 2. 2.1 provides an explicit relationship between
the coefficients aj j : l, 2, . . . ,n of (2. 2. 8) where r : n and coefficients
aij of A in (2.2. 9). However, these results are only applicable when

it is known that CAC”l has the form of a companion matrix.

15

From a practical standpoint, the existence Thm. 2. 2.1
might at first be considered to be of academic interest only. How-
ever, in the proof of that theorem, a new method for converting a
system of n equation (2. 2.1) into a higher-order differential
equation by transform techniques is given, i.e. X : C_1Y (Thm. 22]).
Before the transform technique can be applied the practical problem
of determining a transformation of the type X : C-lY which converts
the system into an r-order differential equation, must be solved.

A transformation of the form

_ , ..
x, -1 .511)
: G
x X(n—l)
- n- L“ _.

 

 

 

 

where G is a nonsingular matrix, is determined in the following:

Consider the system X : AX (2. 2. l) partitioned as

.—

X A A X

(1 l 11 12 1
df = (2. 2.10)
x11 A21 ann Xn
, _ _ ‘ ‘ . y .1.
where Xl — [XI’XZ’ . . . ’xn-l]° Take 11 lsuccessne derivatn es
of A X : —a x + x the last row in (2. 2.10) eliminate each
21 l nn n n
time the first derivative of the vector Xl by means of X1: All Xl +
Alzxn the first row in (2. 2. 10). This formulation results in the

following matrix form:

l6

 

 

 

 

 

 

 

 

V ' F ' _ '1 ‘1
A21 x1) aim 1 0 0 o [xn
A A x -A A -a l 0 0 X(l)
21 ll 2 = 21 12 nn n
3 n—2 ' _ n—3 4 n-5 2(n-1)
AZIAll [Kn—g [AllAllAlz $219111 A12 "P21A11A12m ‘ann lj __"n J

This last equation can be written in symbolic form as

13xl : de (2.2.11)

where the 1 row of (2.2.11) is

1.1 _ 1—2 0 (1—2) (1—1) (1)
A'21A11 X1 ‘ “A21A11 A12Xn'” ’ ° ‘ A21“"11‘A‘12Xn ‘ annxn + xn

1: 1,2, . . . ,n-l. By attaching ones and zeros to (2.2. 11) form

0 1 “x U x)
‘ 1 = [ ‘1‘, (2.2.12)
B 0 x P
n
whereU:[l,0,0,....O].

If B”‘1 exists, the coefficient matrix on the left
side of (2. 2. 12) is nonsingular. The coefficient matrix on the right

side of (2.2.12)

is nonsingular, since matrix L is lower triangular with ones on

the main diagonal, det (L) : 1. Let

G = = L“ B (2.2.13)

17

If the aij entries of matrix A of the system (2. 2. l) satisfy the
condition det (B) f 0 then Blnll exists in (2. 2. l3) and the matrix
C defined by (2. 2.13) is nonsingular. This result is stated by

the following le mma:

Lemma 2. 2.3: If, corresponding to the normal system

 

(2. 2.1) the aij entries of the matrix A satisfy det (B) i 0, where

the i—row of matrix B is

 

[a a . . . a ] a a a 7 1-1
n1 112 n,n-l ll 12 l,n-l
i
[Sn-1,1 an-l,2 ° ' ' an-l,n-l
1: 1:2: ' :n‘l:
then there exists a nonsingular matrix C such that
X : OX (2. 2.14)

d

(1) (n—l)
., xn ].

I ..
where Xd — [xn,xn , ..

Lemma 2. 2. 3 is a significant result of this section
with respect to application. In the technique for constructing the
nonsingular transformation (2. 2. 14) it is shown that the condition
det (B) 3! 0, on the aij entries of the matrix A, must be satisfied for
the existence of the transformation matrix C. Theorem 2. 2. 3
establishes that the transformation (2. 2. l4) converts the normal system
into an n-order differential equation which is associated with the
system. The hypothesis of the theorem necessitates the same
conditions on the aij entries of matrix A in the system (2. 2.1) as

specified by the hypothesis of Lemma 2. 2. 3.

18

Theorem 2. 2. 32 If det (B) )4 0, where matrix B is

 

defined in the hypothesis of Lemma 2. 2. 3, then there exists an n-

order differential equation associated with the system (2. 2. 1),

Proof: For X : AX, there exists a nonsingular matrix
C such that Xd 2- GX by Lemma 2. 2. 3. Substituting X : G-le into
(2. 2.1) results in

-1
Xd — GAG Xd (2.2.15)

The last row of (2. 2.15) is an n-order differential equation associated
with (2. 2.1). For if Fm is a fundamental matrix of (2. 2.1), by
Lemma 2. 2.1, GFm is a fundamental matrix of (2. 2.15). By (2. 2.14),
the entries in the first row of GFm are solutions on I of the n-order
differential equation in (2. 2. 15). In addition, the entries in the j—row
of GFm, j = 2, 3, . . . ,n are the j-l derivatives of the entries in the
first row. Therefore, the determinant of GFm is the Wronskian of
solutions on I of the n-order differential equation in (2. 2.15) and since
G and Fm are nonsingular, GFm is nonsingular. It follows from
Thm. A. 2 that the entries in the first row of (3Fm are also a fund-
amental set for the n-order differential equation in (2. 2.15).

Suppose yl,‘y2, ,yn is a fundamental set for the n—order

differential equation of (2. 2. 15). Then the matrix YJ! : [y., y(.l), ...,yfl-D],

J J

j : l, 2, ,n is composed of n linearly independent vector solutions of

(2. 2. 15). Since substitution of Xd: GX of Lemma 2. 2. 3 into (2. 2.15)

results in (2. 2.1) the vectors G-le, j : 1,2, . . . ,n are n linearly in-

dependent solutions on I of (2. 2.1). The theorem follows.

 

(1.-I55

 

19

The proof of Thm. 2. 2. 3 suggests a new method of
deriving n—order differential equations. If the coefficients aij of
matrix A (2. 2. l) satisfy the condition det (B) if 0 of Lemma 2. 2. 3,
then an n-order equation can be formed by substituting X : C:- le
into X : AX. The n-order equation is the last row of the resulting
system of equations (2.2. 15). Note that the coefficients appearing

in the n-order equation have not been given explicitly in terms of

aij entries of matrix A.

Techniques for obtaining the higher order differential
equation presented by Moulton [5, p. 6] and others [4, p.126] do not
include the nonsingular transformation, X : G-lxd, nor do other
methods restrict the final equation to one of n-order. Moulton,

for example, presents a system of three equations which converts

into a second order equation in any variable.

Corollary 2. 2. 3 has been included to illustrate
a class of system models X : AX which can be converted into an
n-order differential equation, that is associated with the system.
Note, in the proof of the corollary, how the coefficients aij of
matrix A are subjected to the test of det (B) 14 0 (Lemma 2. 2. 3).
This test establishes the existence of an n-order homogeneous ‘

differential equation associated with the system model.

Corollary 2. 2. 3: Suppose matrix A of the system

 

(2.2.1) is .- _
all 0 . .o 0 aln
0
A Z 0 5:122 61211
O O O a
n—ln
anl anZ . . . ann-l ann

 

 

20

where aii distinct i = 1,2, . . . , n and anj 1‘ O for j : 1,2, . . . ,n.
Then there exists an n-order homogeneous differential equation

associated with the system (2. 2.1).

Proof: That det (B) )f 0 follows from the fact that

 

the i-row of matrix B is

—,i-l
bi — [anl anZ ann-l] C111 0 0 W
i—l
0 a22 . O
i-l
O O an-ln-l

 

 

and the refore

det(B):det[b' b' b']:

a a— ...a
l’ 2’ ’ n nan nn-l[
l
I
1..
(J.

l l l

‘11 322'” an-ln-l

 

i
[a

I'D-2 ail—2 an-2
11 22 cn-ln—l
: anl ar12 . . . ann-l |i>j| (aii - ajj) which by hypothes1s 18

not zero. Therefore the corollary follows.

2. 3 Additional Properties of Associate Differential Equations

 

The mathematical properties established in Sec. 2.2
are applied in the development of mathematical techniques for
relating the aij entries in matrix A of (2.1.1) to a specified solution
of (2. 1.2), associated with the system. The techniques developed
here relate the aij entries in matrix A of the system (2.1.1) to

one specified component xi of the vector X of the system.

21

The relationship between the aij entries of matrix A

in the system (2. 3. 2) and a specified solution to the r—order dif-

ferential equation (2.3.1), associated with the system, is established

from the solution properties specified in Def.. 2.2. 1.

Theorem 2. 3.1: If yl,y2, . . . ,yr is a fundamental set for

dr r 1 r
__>:1_ ... 2: a. —__—.‘ 1,] (2.3.1)
dt j:1 3 dt 3
and (2. 3.1) is associated with
X : AX (2.3.2)

“here A : [an] , X' : [Xl’XZ’ . . . ,x ]. Then there exists an

1.1
explicit relation between the r2 entries of A and the entries of C

and Yj’ j : 1,2,. . . ,r, (Def. 2.2.1 for notation).

Since (2. 3.1) is associated with (2. 3. 2) the

vectors C-le where Y' : [0,. .. ,y’j»Y§l).. . . , ’(jr—l),0,. . . ,0].

Proof:

 

j : 1,2, . , r are r linearly independent solutions on I of (2. 3. 2).

, Y ] and F be the (nonsingular) submatrix
r r

of F containing the columns [y., yll), . . .
m J J

> = Y
LctF [1,Y2,...
’ (gr-1)]: j: 1,2,...,1'.

Since the matrix C-lFm satisfies (2.3. 2),

Multiplying the above equation on the right by [0, Fltl, 0] results

in a system of equations in the following form:

22

where Cil is the set of columns of C.1 corresponding to the

position of the entries in F (F ) in F (F ). Since the columns
r r m m

of C11 are linearly independent, there exists a nonsingular sub-

matrix, C;l, of C11 or order r. Multiplication on the right by

Cr results in a system of equations which contains a subset of

equations in the form

A = (2 Fr}? c (2.3.3)

Since Ar is square and therefore contains rZ entries of A, the

theorem follows .

Corollary 2. 3.1: With the same hypothesis and r : n.

 

A 2 c F. F c (2.3.4)

Proof: This is a direct consequence of Thm. 2. 3.1.

 

 

The matrix Fm as specified in (2. 3. 4) is

 

 

yl yz o . . yn j
(l) (l) (1)
F ; yl Y2 . . . yn
In ....... (Z. 3. 5)
(11—1) (n—l) . . . (n—l)
It is proved in the following corollary that FmF-rrli

has the special form of a companion matrix.

23

Corollary 2. 3. 2: With the same hypothesis and r : n

the F F-1
m m

 

of Cor. 2. 3.1 is

fo 1 o o
. -1 o o 1 ... o
Fmsz ....... (a3t)
o o 0 1
bn bn-l bn-Z bl_

 

 

Proof: Let T represent the companion matrix on the

right of (2. 3. 6). Consider the matrix product T Fm. If the n

equations
(1) y(n—l) _ '(n)
bnyl+bn-ly1 +... +blyl - yl
(1) (11-1) _ (11) ‘
bn y‘2 + bn_ly2 + . . . + blyZ — y‘2
.(1) ,(11-1) _ ',(n)
bnyn+bn~1yn + +blyn — yn

are satisfied then T F : F
m m

The above system of n equations in matrix form is

 

 

 

 

 

 

_ _ _
y1 Sf1 "' YYPDW bn T Vii
. (n-D (D)
y2 y2 y2 baa. 2 Y2 (2 3 7)
....... (n-l) : 6])
[in yn ° ' ° yn bl Lyn
_ 2 1 1

From Thm. A. 2, Fm is nonsingular. Since the matrix on the

left of (2. 3.7) is F51, the lemma follows.

24

Corollary 2. 3.1 establishes a relationship (2. 3.4)
for determining the aij entries of the matrix A in (2. 3. 2) in terms
of one specified component xi(t) of the vector X(t) of (2. 3. 2).
When Thm. 2.2.3 is applied to Cor. 2. 3.1 it is determined that if
the aij entries in matrix A satisfy det (B) l 0 then the vector com-
ponent xn(t) of X(t) in the system (2. 3. 2) can be specified as consisting

of n linearly independent parts. This specification in turn restricts

the a.lj entries of matrix A by

Azo'lir F‘lo (2.3.8)
m

m

Equation (2. 3. 8) is a principal result of this section and is referred
to as a "design equation" when applied to the design of linear

oscillators in Chapter V.

The results of Cor. 2. 3.1 are extended one step
further in Cor. 2. 3. 2 by proving the matrix product .FmFr-nl in
(2. 3.4) or (2. 3. 8) to have the form of a companion matrix (2. 3. 6).
An important consequence of this, which is applied in Thm. 2. 3. 2,
is that the matrix products CACE1 (2. 3. 4) and GAG-1(2. 2.15),
(2. 3. 8) have the form of a companion matrix. This implies that

the characteristic polynomial of matrix A, det [A - XI] must be equal

to the minimum polynomial [10, p. 149] of a matrix A.

Gantmacher [10, p. 159] shows that the matrix C in
the similarity transformation CAC”1 , which produces the companion
matrix, is not unique. Faddeeva [11, p. 201] presents a method
developed by Danilevsky which brings a matrix A into companion

matrix form by means of (11-1) similarity transformations. The

25

primary difference in the method for obtaining the companion matrix
in this thesis and other methods is that here it is obtained under
specified conditions det (B) i 0 on the aij entries of matrix A by

a particular matrix product GAG-l, which has the desired form of
a companion matrix. This is not the case in the other methods
which have beenestablished for the primary purpose of bringing

the characteristic determinant det [A - X I ] of a matrix A into
polynomial form. What is even more significant in the development
of this thesis is the presentation of the transform in the form

X : G-l Xd (Lemma 2. 2. 3) which links the solution of an n-order
differential equation (2. 3. 9) to the solution of the system of dif-

ferential equatiom(2. 3. 2). The result of GAG.1 having the form of a

companion matrix is applied in the following theorem.

Theorem 2. 3. 2: If det (B) 75 0, where matrix B is

 

defined in the hypothesis of Lemma 2. 2. 3, then

:3
:5
:3
I
H

 

(2.3.9)

——x:2‘,a.
nn -1

dt 1
Where aj is (-1)j+l times the sum of the principal minors of order
j of A, is associated with

x : AX (2.3.10)

1.

where A : [aij] and X' : [xl,x2,. . . ,xn

Proof: Theorem follows from Cor. 2. 3. 2 and

Thm. 2.2.3.

26

 

Theorem 2. 3. 2 provides a simple, yet effective
procedure for establishing a mathematical relationship between
the aij entries of the matrix A in the system (2. 3.10) and the
aJ. entries in the n-order equation (2. 3. 9). If the matrix A is
given and det (B) 34 0, or if matrix A contains arbitrary entries \
and det (B) i 0 is specified, then the mathematical relationships
are given in the theorem. The results of Thm. 2. 3. 2 provides

a new tool for system design in a later section.

Equation (2. 3. 8) is the result of specifying only one
component xn in the vector X of the system (2. 3.10). On the
other hand, Thm. 2. 3. 2 relates only the aij entries of the matrix
A in the system (2. 3.10) to the a]. entries in the n—order equation
.(2. 3. 9). The additional step of relating the aij entries to the

specification on xn is made possible by the following theorem.

Theorem 2. 3. 32 Given the n—order differenial

 

equation

 

n n n-j
97x: :3 a. dn_ x (2.3.11)
dt j:1 3 dt 3
k kit
(1) If a solution on I of (2.3.11) is y(t) : f.) Pm _l(t) e (B.2)
1:1 1

then aj, j : 1,2, . . . ,n is given by (B. 1.1), (B. 1.2) a.i : -bj
for m. : 1 for all i.
1 r kit
(2) If a solution on I of (2. 3.11) is y(t) = E cie , where cifio,
1:1
and 1‘1 )4 0 and distinct, thenr coefficients of (2. 3.11) are given

by (B. 1.4) where a‘j : -bj.

27
m at
(3) If a solution of I of (2. 3.11) is y(t) : Z) cie 1 cos (wit +61) and
1:1
2m : r, then r coefficients of (2. 3.11) are given by (B. 1.4)
where a. : -b..
J J

Proof: The theorem follows from the general form

of solution given in (B. 2) .

 

Theorem 2. 3. 3 establishes parameter-solution
relationships between the coefficients aj of (2. 3. 11) and a specified
solution. In (2) it is interesting to note (Thm. B. 1. 2) that if a
solution with r linearly independent parts is specified, r coefficients
of the n—order differential equation can be expressed in terms of the

remaining n -r coefficients .

Theorems 2. 3. 2 and 2. 3. 3 are sufficient to interrelate
the aij entries of the matrix A of the normal system (2. 3.10) and
the component xn(t) of the vector X(t) in the system (2.3.10).

These mathematical relations are referred to as "design equations"

when applied to system design in Chapter V.

III. PROPERTIES OF SYSTEMS OF NONHOMOGENEOUS

 

DIFFERENTIAL EQUATIONS AND ASSOCIATED

 

HIGHER-ORDER DIFFERENTIAL EQUATIONS

 

3 . 1 Introduction

 

Mathematical properties parallel to those in Chapter II
are developed for relating the solution of the nonhomogeneous
system

x -.- AX +Q(t) (3.1.1)

, 1 _ 1 _.
where X _ [Xl’XZ’ . . . ,xn], Q (t) _ [ql(t),q2(t), . . . ,qn(t)],
A : [aij] to the solution of the r—order(rf_n) differential equation

r-i
d
i r-i

dt

 

 

y +F(t) (3.1.2)

For instance. the problem of determining a transformation of

the form

x : c'1[Ys - L‘1H(t)] (3.1.3)

where X' : [Xl’XZ’ . . . ’Xn]’ Y‘S : [0, 0, . . . ,0, y,y(l), . . . ,y(r-l),0, 0,...,0],
H(t) is a vector function of t, and C and L are nonsingular matrices,
which links the solution of the normal system (3. l. l) to the solution
of the r-order equation (3. l. 2), is encountered. The solution of this
problem will allow the system designer to determine the initial

condition of the physical system in terms of one component Xi(t)’

Of the system solution X(t).

28

29

In contrast to the problems considered in Chapter II,
a new problem that arises in this section is that of formulating the
vector Q(t), of the normal system (3.1.1), in terms of one component
xi of the vector solution, X. These parameter—solution relation-

ships are illustrated in the design of amplifiers in Chapter V.

3.2 Systems of First-Order NonhomOgeneous Differential

 

Equations and Associated Higher-Order Differential

 

Equations

Properties parallel to those in Chapter II are developed
here for the nonhomogeneous system (3. 2. 1). For instance in
Thm. 3. 2. 1 it is proved by applying a transformation of the form
(3. 1. 3) to a nonhomogeneous system, that there exists a set of s
differential equations, 1:s_<_n, of the form (3. 2. 2) "associated"
(Def. 3. 2.1) with the system. In Thm.3. 2. 2 conditions on the aij
entries of matrix A in the normal system are given so that there
exists a differential equation of n-order associated with the system.
In the proof of these results a technique for formulating a trans-

formation of the form (3.1. 3) is given (Lemma 3. 2. 2).

Definition 3. 2.1 provides a concise description of the
mathematical properties existing between the normal system
and the r-order equation associated with the system. These

properties are clarified in the theorems of this section.

30

Definition 3. 2.1: Consider the system of n non-

 

homogeneous equations,
X :AX +Q(t) (3.2.1)

where A :[aij1’ X' : [x1.x2, - . . ,Xn], Q'(t) =[ql(t)..-.qn(t)].

qi(n)(t), i : l, 2, . . . ,n is continuous for all t on the open interval I

defined by I : [t: t1 < t < t2], where t1 and t2 are constants.

An r-order nonhomogeneous differential equation

I'

dy

r .
dt 1

+ F(t) (3.2.2)

 

 

r
"E a.

l
is associated with (3. 2.1) if:

(a) The homogeneous part of (3. 2. 2) is associated

(with nonsingular matrices C and D and row i) with the homogeneous

part of (3. 2. l) and
(b)l.For X'(t) : [xl(t),x2(t), . . . ,xn(t)] the solution of
(3. 2.1) on I such that X(to) : 0, t0 on I, then row i of CX(t) is the

solution of (3. 2. 2.) on I which is zero at to,
2. For y(t) the solution of (3. 2. 2) on I such that

Y(J)(t0) = 0. forj = 0,1,... ,r-l, tO on I, and

l r-i--1pr_iJrl r—l r
. D f . t + 23 P. D f .t ,
J (>,_ ,1) J()r_3()

1—

0 j:0
(3. 2. 3)
where

_ .1 1‘1 '
PfiD)_D +lk,k-1D +lk,k_ZD +...+1k’k_j,

31

. j -
d
D] ___.__,., 13:1) 3“ Zplip) -

-1 -l .
then C [Ys(t) - L H(t)] 18 the solution of (3.2.1) which, when

evaluated at to, is the zero vector, where L.1 : [1..] and
1
1 l .-
Y (t) : [0,0, . . . ,0,y(t),)()(t) y(rb(t),0, . . . ,0],
bY—d

ate): [0,0, ,.,o,g(n,1[”(n +£Zn),
l
fl(r-Z)(t) + f (r'3)(t) +

2 +fr_l(t),0,...,0].

Definition 3. 2- Z? An r—order homogeneous differential

 

equation

'1
’1

 

Cl ._ ' dr-l

dt 1 1 1 dtr-l

is associated with (3. 2.1) if (a) and (b) of Def. 3. 2.1 are satisfied.

The mathemtical properties specified in Def. 3. 2.1
are first encountered in Lemma 3. 2.1 which follows. The lemma
presents a normal system of the form (3. 2.1), with the coefficient
matrix in the form of a companion matrix, that converts into an n—order

differential equation associated with the system.

32

Lemma 3. 2. 1: Suppose matrix A of the system(3. 2.1) is

 

 

To 1 0 07
0 0 l 0
A z . (3.2.4)
0 0 0 1
an an—l an-2 al

 

Then there exists an n-order differential equation, associated with

the system (3. 2.1).

Proof: Determine the n-l successive derivatives of the
first row of (3. 2.1) eliminating each time from the right hand
,x1 by means of the last

 

the first derivatives of x2,x3, . . .

 

 

side,
n-l original equations. This process results in
X1 : X + Ql(t) (3.2.5)
, 1_. (l) (fl-l) I-
where X1_[Xl’xl ,... ,xl ], X - [xl,x2,... 'Xn] and
, _ (1) (11-2) (11-3)
010:) ~ [0. qltt). ql(t) + (12199”wa (t) + C12 (0 + + qn_1(t)]
and
dn n dn-j n—l
n x1: '23 a. 11..xl+ E P.(D) qn_.(t) (3.2.6)
(it j:1 Jdt J j—_-0 J 3
where P.(D) :DJ -a1D‘]- -a DJ.2 -...-a.,j 1,2,. ,n-l,
1 . Z 1
dJ

PO(D) : 1 and DJ :
dt‘]
That the homogeneous part of (3. 2. 6) is associated with

the homogeneous part of (3. 2.1), with C and D unit matrices and i :1

follows by an argument similar to that used in the proof of Thm. 2. 2.1.

33

If X'(t) = [xl(t),x2(t),...,xn(t)] is the solution of (3.2. 1)
on I such that X(to) : 0, t0 on I, then by the method of construction of
(3. 2. 5) and (3. 2. 6), xl(t), the first entry of X(t), is the solution on
I of (3. 2. 6) such that xl(t0) : 0. Therefore if (3. 2. 6) is homogeneous,

the lemma follows.

Suppose (3. 2. 6) is nonhomogeneous and let y(t) be the

solution of I of (3. 2. 6) such that y(J)(tO) : 0 for j : 0, 1,2, . . . ,n-l

’

tO on I. It has been shown [4, p. 134] that such a solution exists.

0'1 n-l
For F(t) : Z) P.(D) q .(t), form F(t) 2 E P.(D) f .(t), which
j:0 J “'21 j30 J 11".]
is a special case of (3.2.3) with 1.. : 0 for i i j and 1.. = l for i :
k-2 . 1’] 11
1,2,...,n, such that z Djfil. (t ) = o, t onIandk = 2,3,...,n.
j:0 ~-j o o
This can always be done, since if fi(t0) : 0, 121,2, . . . ,n-1,
n-1
F(t) : fn(t) : 2'3 Pj(D) qn_j(t). Consider the transformation of
i=0
variables xl(t) : yl(t) and

Y(t) = AY(t) + 131(1) (3.2.7)

where Y'(t) : [yl(t),y2(t), . . . ,yn(t)], F'l(t) : [fl(t),f2(t), . . . ,fn(t)]
and A is given by (3. 2.4). The solution of (3.2. 7) is Y(t) : Ys(t) - Fll(t)
where YS:(t) : [y(t),y(lltl ..... ,y(n_l)(t)] and

1 __ (l) - (II-Z)
F11“) .. [O,£l(t),f (t) +12(t),...,fl
and Y(to) : Ys(to) - Fll(to) : 0 for to on I since Ys(t0) : 0 and

(t) + fg1'3)(t) +. . . + fn_l(t)]

Fll : 0 for to on I. This implies the lemma.

Corollary 3. 2.1: With the same hypothesis and if

 

n-l
1; Pj(D) qn-j(t) : 0 then there exists an n-order homogeneous differential
1:0

equation associated with (3. 2.1).

34

Proof: This follows from the method of formulating

(3. 2. 6) in the proof of Lemma 3. 2.1.

A review of the salient features in the proof of Lemma 3. 2.1
will lay the foundation for the theorems which follow. First, a
nonsingular transformation of the form (3.1. 3) is determined
(3. 2. 5 in Lemma 3.2.1). This transformation relates the solution
of the system to the solution of the higher-order differential

equation (3. 2. 6) .

Property (a) of Def. 3.2.1 is established in the same

manner in which the results of Thm. 2. 2.1 were established.

In Property (b)1 of Def. 3. 2.1, the first row in the vector
CX is the solution of the higher-order equation (3. 2. 6). In the
lemma, matrix C is the unit matrix and the first entry in the

vector Ql(t) is zero.

To satisfy Property (b)2 of Def. 3. 2.1, it must be
pointed out that there exists a solution of the higher-order equation
with the property y(j)(to) : 0,j : 0,1,2, . . . ,n-l. Next, it must be
demonstrated that the nonhomogeneous part, F(t). of the higher -
order equations, can be put into the form specified in (3. 2. 3) in
such a way that it satisfies the conditions at tO specified in (3. 2. 3).
It is shown in Lemma 3. 2. 1 that F(t) can always be put into the
desired form to meet the specified conditions t : to. The reason for
this last condition will become clear in the proof of Thm. 3. 3.1

which follows late r .

35

It may be noted at this point the condition at tO on F(t)
forces the vector L-ll-i(t) in the transformation (3.1.3) to be zero

at to. In Lemma 3. 2.1 this requires F to) to be zero. Similar

11(
operations are applied in Thm. 3.2. l which follows to prove the

existence of a set of s differential equations,1< s< n, of order ri

associated with the system.

Theorem 3. 2.1: There exists a set of s differential

 

equations, l< s< n,of order ri, 1: 1,2, . .. ,5, associated with the
5
system (3.2.1) such that :7 r1: n.
i=1

Proof: Consider the transformation X : C—IY on

 

(3. 2.1) which is used in the proof of Thm. 2. 2.1. For this case

the transformed system of equations is

i' = CAc'lY + con) (3.2.8)

and
Y1: BiYi + Fin) (3.2.9)

where Bi,i : 1,2, . . . ,s, is of the form of (3.2.4). By Lemma 3. 2.1
there exists an ri—order differential equation (3. 2. 6), n : ri,
associated (with matrices C and D and row i) with (3. 2. 9). That the
homogeneous part of this differential equation is associated with the

homogeneous part of (3. 2.1) follows by an argument similar to that

of the proof of Thm..2.2. 1.

If X(t) is the solution of (3. 2.1) such that X(to) : 0, t0 on
I, then CX(t) is the solution of (3. 2. 8) which is zero at to.

The vector [yil(t)ay12(t)f . .inri(t)]' whgrcl: the component yij(t),
1: 1,2,. .. , s, j: 1,2,... ,ri, is the E rp + j entry in CX(t),
p=l

36

is the solution of (3. 2.8) which when evaluated at to’to on I, is

zero. Therefore yi t) is the solution of the ri-order differential

l(
equat1on, (3.2.6), n : ri, for wh1ch yil(t0) :: 0.

If (3. 2. 6) , n : ri, is nonhomogeneous, then by an
argument similar to that of the proof of Lemma 3.2.1 (see proof
for notation), the solution of (3. 2.9) is Y(t) : Ys(t) - F(t) which
is zero at to, tO on I. Appending zeros to the vectors expressing
this solution of (3. 2. 9) results in the solution of (3. 2. 8) which is
zero at to. Since the solution of (3. 2.1) is X : C-lY, the ri-order
differential equation established by Lemma 3. 2. l is associated with
(3. 2.1). The theorem follows since the above argument applies

5

for all i, i: 1,2,...,s and E ri: n.
irl

 

In Theorem 3. 2.1 it is established that the normal system
(3. 2.1) converts into a set of s differential equations, l_<_s_<_n,of
order ri. This is an extension of results found by Murray and Miller
[4, p. 129] and others [5, p. 6]. In addition to this, from a
practical viewpoint, the proof of the existence theorem affords a
new method of determining a solution of a system in terms of a
solution to a higher-order differential equation, i. e. X : C-lY.

The problem of formulating a nonsingular transformation, of the

form applied in the theorem is the subject of the following discussion:

Consider the normal sysaem X : AX + 0(1), partitioned as

d
21'? - + (3.2.10)

 

 

 

 

 

 

As in the proof of Lemma 2.2.3, take n-l successive derivatives
of the last row in (3. 2.10), eliminate each time the first derivative of
the vector X by means of the first row in (3. 2.10). This formulation

1

results in the following (n-l) relations:

‘1 “its???“ “ “*r—
f ..A

 

 

 

 

 

 

 

 

—A l - v _ 1 o o 01 P '
21 X1 -ann Xn
= - ‘ - o . (1)
A21AM X2 ; A21A1H2 ann l O xn +
1011-2 I -5 :(n-l)
- - 1
A2113111 xn—l A213111A 12 AZIAllA 12 AzfdllAlz “m Xn 1
_. ._ _ L— - l_
to
-qn(t)
-3 n-3
-q:1a(t)+ AZlA‘il (2(1“ )+ ..+ AZlAll Ql(t)

 

 

This last equation can be written in symbolic form as

13x1: de — 02(1) (3.2.11)

where the 1 row of (3. 2.11) is

i- o M (1-1) (1)
AZlAllX :AZIAilzlA 12X 11 ° ° ° -A21A11A12Xn -annxn + X11

+ (1:1—1)“) + AzlAleli-2)(t) + ... + AZIAl 2

38

and i : 1,2, . . . ,n-l. Let (3. 2.11) be bordered with ones and zeros

to form

0 1 X1 U [Xd] - 0
: (3.2.12)
B 0 x
n

1
P- [‘QZ(t)_
where U :- [l,0,0,. .. ,0].

-l . . . . .
If B ex1sts, the coeff1c1ent matrix on the left Slde of

(3. 2.12) is nonsingular. The coefficient matrix

also defined in Lemma 2. 2. 3 is nonsingular, since matrix L

is lower triangular with ones on the main diagonal, det (L) a l.

 

Let
-1
U] 0 1'[ 1
_ )_ -
G~ —L Bll (3.2.13)
p B o]
and
O
Qd(t)=
sz

If the aij entreis of matrix A of (3. 2.1) satisfy the condition det (B) 15 0
then Bi; exists and the matrix G defined by (3. 2.13) is nonsingular.

These results are stated by Lemma 3. 2. 2 which follows.

Lemma 3.2. 2: If corresponding to (3.2. l), det(B) )4 0,

 

then there exists a vector Qd(t) and nonsingular matrices G and L

such that

X = GX+L'le(t) (3.2.14)

where matrix B is defined in the hypothesis of Lemma 2. 2. 3

an“)... ..(n-l) 1.

and X' : [x
n

d

n ' '

Lemma 3. 2. 2 is one of the significant results of
this section. The lemma is, in a sense, an existence theorem.
That is, if the aij’entries of matrix A in the normal system (3. 2.1)
satisfy the condition det (B) ,4 0 then there exists a nonsingular
transformation (3. 2. l4). Identical conditions on the aij entries in
the homogeneous systems were found (Lemma 2. 2. 3) for the
transformation (2. 2.14) to exist. Theorem 3. 2. 2 shows that the
transformation (3. 2. 14) does convert the normal system into an

n-order differential equation which is associated with the system.

Theorem 3. 2. 22 If det (B) )4 0, where matrix B is

 

defined in the hypothesis of Lemma 2. 2. 3, then there exists an

n-order differential equation associated with the system (3. 2.1).

Proof? Substituting the transformation (3. 2.14) into the system

(3.2.1) results in

. -1 -1 -1 _1.
Xd : GAG Xd - GAG L Qd(t) + L Qd(t) + GQ(t) (3.2.15)
Whe[:’lo (t +GQt ]' —- [f (t f(l(t) +£ (t 1"”) + 18“” +
d) (l ‘ 11 )’11 12 )"°"11(° 12

+ flnm] [151]:

 

40

+ f12(t),... ,fma(t)+f(n23)(t)+... H (t)]

, 1)
Q(1(tlo'f((911t)'1(11 11 l,n-l

_ ‘ i-2 ._
and fll(t ) — qn(t), fli(t) — AZlAll Ql(t), 1 — 2,3,. . . ,n. The last

row of (3.2.15) is,

 

n n n—i n—l n- -i- l

9—n—xn — 2 .4i d mi xn - >3 ai z P‘?’i+l(D) 11 _._.(t) +

dt 1:1 dt 1:1 3'20 “ 1 J [
n-l p“ LU
£0 J.(D) £1,11_J.(t) (3.2.16)

”fi:.'7m-.---.- ---—~.—:
‘

an n-order differential equation associated with (3. 2. l), where PfiD) is
given after (3. 2. 3) and L”1 = [lij1' F01; by an argument similar
to that of the proof of Thm. 2. 2. 3, the homogeneous part of (3. 2. 16)

is associated (with matrices G and row one) with the homogeneous

part of (3. 2. 1)

If X'(t) : [xl(t),xz(t), . . . ,xn(t)] is the solution of (3. 2. 1)
such that X(to) : 0, t0 on I, then the corresponding solution of (3. 2.15)
is given by (3. 2.14). The first entry of Xd(t) which is the first
entry of CX(t) (since L”1 is lower triangular) is the solution of

(3. 2. 15) which is zero at to. Therefore if (3. 2.16) is homogeneous,

the the orem follows .

For the case of (3. 2.15) nonhomogeneous the method of the

last part of the proof of Lemma 3. 2.1 is used. That is for

n-2 n—i-2 . A‘ni'J 2 n-l n+1-i
F(t)=- :a,: P“ “111mm,, elm-La .P__nli<D >q<t>
1:1 j:0 1:1
'ZPIRD)A21A1AHJZQ + P: _1(D) qn(t) (3.2.17)

+j:o

41

n-l n-i-l -i+l n-l n
form F(t) = - z a. 2 P“. (D) f ..(t) + z P.(D) f .(t) such
i=1 1 j=0 J n-l-J j:0 J n-j
k—2
that 23 PhD) f .(t ) = 0 for t on I, k : 2,3,. . . ,n. Consider
j:0 j k-l-j o o

the transformation of variables xn(t) : y(t) and

l

l13‘1c2du) + L" Fl(t) (3.2.18)

Y(t) = GAG"1Y(t) - GAG"

n~l)

where Y'(t) [y(t), y(1)(t),..- ,y( m]. Dam = [0.£(t).£,”’(t) + f2(t).

[fin-2)“) + ...+ fn_l(t)], Fl'(t): [fl(t),f(l)(t)+fz(t),... ,il(n’1)(t) +

. + fn(t)]. Substituting (3.2.14) into (3. 2.18) results in (3. 2. l)

sinceQ'(t) :' [f1(t),f2(t), ,fn(t)] (GI—1L-l

)' . Therefore if y(t) is
the solution on I where y(to) : 0 and y(J)(tO) : 0, j : 1,2, . . . ,n--l,tO
l

od(t)]

where Xci(t) : [y(t),y(l)(t), . . . , y(n-1)(t).] such that X'(to) : O.

on 1, then the solution of (3.2. 1) is X(t) —.— G'l[xd(t) - 1.’

The theorem follows .

The determination and application of the nonsingular
transformation of Lemma 3. 2. 2 and Thm,3. 2. 2 are unique to this
thesis. The technique used in the proof of Thm. 3. 2. 2 offers not
only a new method of formulating an n-order differential equation,
but even more important a closed form relationship X : G-le-L-IQAtH
linking the solution of an n-order equation (3. 2. 16) to the solution of
the normal system (3. 2.1). Methods of obtaining higher-order
equations presented by Moulton [5, p. 6] and others [4, p. 126] do
not consider a transformation of the above type (3. 2.14) and as a
result do not restrict the final equation to one of n-order. The

restriction of the final equation to one of n-order and the transformation

42

in the special form (3. 2.14) are important features in the design of

electrical networks proposed in a later section.

A review of the main features of Thm. 3. 2. 2 shows
that if the: aij entries of matrix A in the normal system (3. 2.1)
satisfy the conditions det (B) :4 0, then the system converts into the
n-order differential equation (3. 2.16). This n-order equation, which

is associated with the system (3. 2.1), is the result of long and

 

difficult derivations, i.e. the last row of (3. 2.15). However, now

‘q..;?£;':fi:.-al tuna: ADP

that this derivation has been sucessfully performed for the general
case it is no longer necessary to go through the complete process

of substituting the transformation (3. 2.14) into the system (3. 2.1)

and obtaining the last row of the resulting system, to arrive at the
n—order equation (3.2.16). It is, however, necessary to determine
some of the components of the n—order equation. The ai, i : 1,2, . . . ,n
components in (3. 2.16) are determined by applying Thm. 2. 3. 2

as being (--l)i+l times the sum of the principal minors of order 1

of matrix A in (3. 2.1). Closer examination of the nonhomogeneous part
of the n—order equation (3. 2. 16) shows the only derivation yet to be
made is [...-l : [ lij ], where the matrix L is defined in the results

of Lemma 3. 2. 2. This later derivation is relatively simple since the

matrix L is lower triangular with ones on the main diagonal.

An additional result of Th‘m. 3. 2. 2 is the formulation

of the vector

1 l

I ._ . ' 7 1
Q (t) — [fl(t), f2(t), ..., fn(t)] (G L ) .
of the normal system (3.2.1) in terms of the components fl(t),f(t),...,fn(t)

in the nonhomogeneous part F(t) of the n-order equation (3. 2. 2). The

43

formulation of the vector Q(t) is referred to as a "design equation"

when applied to the design of amplifiers in a later section.

3. 3 Additional Properties of Associated Differential Equations

 

Mathematical properties established in Sec. 3. 2, which

P

relate the solution of the normal system (3. 1. 3) to the solution of

the r-order equation (3.1.2) associated with the system, are

fi-fl-BFAJ-a -.-.3t-‘--
. .
(...! -

applied in this section. In the following development the coefficient

‘11.;

matrix A and the vector Q(t) in the normal system are related to
the solution y(t) and the nonhomogeneous part F(t) of an n-order

differential equation (3.1. 2) associated with the system.

Theorem 3. 3.1: Suppose y1(t), y2(t), . . . ,yn(t) is a

 

fundamental set of the homogeneous part of (3. 3. l) and y(t) is the

solution on I: [t[ > to of

 

dn .n n-j
”7:: z 2: a. n_ x+F(t) (3.3.1)
dt 32:1 3 dt 3

such that y(J)(tO) : 0, t0 on I, j : 0,1,. . . ,n—1 and (3.3.1) is

associated With

>4
1 I

AX +Q(t) (3.3.2)

then

C-l

>
H

- -1
Fm(t) Fm (t) C

(3. 3. 3)

em = me git-(Fm Y(t))

 

 

 

44

WhereFrnU) :[fij(t)1v fij(t) :ygl-1)(t), i, j : 1,2, . . . ,n and

Y(t) : C"l [Ys(t) - L-l H(t)] (see Def. 3. 2.1 for notation).

Proof: By hypothesis, and Cor. 2.3.1 (3.3.3) of

conclusion follows .

If Fm(t) is a fundamental matrix for X : AX then, by [

Thm. 3.1 [6, p.74]

 

Wr— _“

t
Y(t) me] Fljnlm) Q(s)ds
t
O

t on I, is that solution of (3. 3. 2) satisfying Y(to) : 0. Application

of "Leibnitz Rule" (Thm. A. 4) to

t

d -1 V __ d —1

2175...“) Y(t) ) NEE]: Fm(s)Q(s) ds)
0

results in n relations,
Q(t) — F (t) ——d (F-1(t) Y(t )
~ m dt m )

Since the n-order equation (3. 3.1) is associated with the system

(3. 2.2) then Y(t) : c'1[ 1

Ys(t) - L- H(t)] is the solution of (3. 3.2)

which, is zero when evaluated at t0, t0 onl .

In Theorem 3. 3.1, if y(t) is replaced by xn(t) and
the restriction det (B) i 0 (as defined in the hypothesis of Lemma 2. 2. 3)

is added to the hypothesis of the theorem, then the relationships

A = 6'11?“ F-IG (3.3.4)
m m
d —1

are obtained. The solution of the normal system (3. 3. 2) is given

in this restricted situation i.e. det (B) i 0, by the vector

(t) - L-1H(t)] (3.3.6)

where XC'1(t) : [x (t), x(1)(t), . . . ,x(n-l)(t)], matrices G and L
n n n
are determined as in the discussion proceeding Lemma 3. 2. 2,

and 11(t) is determined as specified in Def. 3. 2.1 where r : n.

The relationship (3. 3.4) has been discussed in Chapter II.

"E 'iilllffifi'”"" ._ ..

IV, ON A CLASS OF SYSTEMS OF NONLINEAR DIFFERENTIAL

 

EQUATIONS AND CORRESPONDING REDUCED DIFFERENTIAL

 

EQUATIONS

 

4. 1 Introduction

 

In the design of tunnel-diode amplifiers and oscillators.
the state models of these systems have taken special forms. By
divorcing the parameters, in these mathematical models from the
parameters associated with a particular physical system, the class

of systems

 

 

 

 

 

 

 

 

 

 

_ .. .. — _. a P ..l r' _

x1 all alp—l a1p+1 an.“ ’f1 {1(Xp) ‘11“)
d I XI“
_._ : a a a a x + f x + t
dt xp pl pp-l pp+l in p+l p( p) qp( )

. i .
- g )

an‘ _anl anp—l dnp+l an“- _xn 2 tin(xp)d [final

or in symbolic form
X : AlXOp + l\(xp) + on) (4.1.1)

results.

The problem of relating the solution of the system (4.1. 1)

to the solution of the n-order differential equation

 

dn n—l C1n-j
dt p j=1 J dt 3 p p

by means of a nonsingular transformation is encountered, as in

Chapters II and III.
46

Wart-h +...—.1 1r—

47
The additional problem of relating a specified solution,
BO + Bl sin(wt+¢), to the parameters in the n—order equation (4.1. 2)
where T(xp.) is a polynomial of m—order is considered. The solution
of this problem will assist in the design of an amplifier or oscillator

when the system contains a nonlinear component whose characteristics

can be approximated by a polynomial of order m.

{1' 'c‘ vamp——

4. 2 Formulation and Solution of Reduced Nonlinear

 

Differential Equations

 

‘ht‘flfi’ - 3

Conditions are given on the normal system (4.1.1)
for the existence of a nonsingular transformation (4. 2. 3). The
transformation, when it exists, reduces the system (4.1. l) to an
n-order "reduced" (Def. 4. 3.1) differential equation (4.1. 2). The
transformation in turn relates the solution of the normal system to
the solution of the n-order reduced differential equation.

First consider the problem of formulating a nonsingular
transformation.

Let the normal system X : AlXOn + N(xn) + Q(t), (4. l. 1)

be partitioned as

d X1 [A11 [X1] F12(Xn) Q1“)
a—t- : ) + +
xn 1! A21 fn(xn) qua)
F (4. 2. 1)

As in the proof of Lemmas 2. 2. 3 and 3. 2. 2, take n-l successive
derivatives of the last row in (4. 2. l) and eliminate the first deriva-
tive of the vector X each time by means of the first row in (4. 2. l).

1

This formulation results in

48

,(i) _ i-1 (i-l) o (i_2) 1.)
"n “A21A11 X1 +fn (xn) +£21A11F12 (an +°'+A21A11F12(Xn)
(1—1) 0 (i—Z) 1-2
+ qn +A21A1101 + +A21All Ql(t)

for i : 1,2,. . . ,n-l. These n-l relations in matrix form are

_ - L‘.‘ r.."—

x = BX +2 (x1, t) (4.2.2)

H
0..
y...
H
H

""tl‘“ :.
“H.-

l n-l
whereXl'd : [xil ),... ,fol )] and XI:[X1’X2"" ’Xn-l]'

Let (4. 2. 2) be bordered with a one and a zero to form

 

 

 

 

 

 

 

x [o 1; x1 _ o P
n . 1

e +
_ded __ B 0_] __ xn - _Zl(xn’t).(

This last equation can be written in symbolic form as

Xd : BllX + Z(Xn, I.)

If in addition, the aij entries in Inatrix A1 of the system (4. l. 1)

satisfy the condition det (B) )4 0 then B11 exists. These results

are. stated by the following lemma.

Lemma 4. 2.12 If corresponding to the system (4.1.1) with

 

p = n, det (B) if 0 where matrix B is defined in the hypothesis of Lemma 2.2.3,
then there exists a vector Z(xp, t) and a nonsingular matrix B11,

such that

 

49

xd :BllX + Z(xp. t) (4.2.3)
where X' : [x ,x(l),. .. , X(n-l) ] .
d p p p

Lemma 4. 2.1 states that if the aij entries of matrix Al
in the normal system (4.1.1) satisfy the midition det (B) ;€ 0 then
there exists a nonsingular transformation (4. 2. 3). The transfor-
mation determined in the lemma is the main part of the Def. 4. 2.1.

The definition supplies an operational technique for determining,and

hence defining,the n—order equation obtained from the normal system.

Definition 4. 2. l: The nonlinear differential equation

 

obtained as the last row of the system of differential equations

—1
11

in (4.1.1) is the reduced differential equation corresponding to

generated by substituting X : B [X - Z(xp, t)] , of Lemma 4.2.1,

(1

(4.1.1).

Consider as an example, the two normal systems in
Theorems 4. 2.1 and 4. 2. 2 that convert to a reduced differential

equation.

Theorem 4. 2.1: If the matrix A1 of the system (4.1.1)

 

with p : n is

50

 

 

f- -!
c 0 O
111 0
0
0 a22 0
A1 =
O O an-ln—Z an—ln--1
Lanl an2 ° ° ' ann -z ann- 1.

where aii distinct i = 1,2, . ,n-1 and anj :5 0 for j : 1,2, . . . ,n—l,
then there exists an n-order reduced differential equation corres-

ponding to (4.1.1).

Proof: The proof is similar to that of Cor. 2. 2. 3

 

and therefore has been omitted.

Theorem 4. 2. 2: If matrix A of the system (4. 1.1)

 

1
with p : l is
-' '1
1 0 o
o 1 0
Al :
o 0 1
En-l an-2 "' a1

 

 

Then there exists an n—order reduced differential equation,

11 11-1 n-j n-1 n-1
-—-—x : 2: a X1+ :3 Pj(D)fn_J(x1)+_Z Pj(D) qn_j(t)

at“ l j:1 Jdtn'J j:0 3:0

 

corresponding to (4.1.1), where Pj(D) : DJ-alDJ-l-a DJ-2-. . . -aj,PO(D) :l

2
d‘]

dtJ

and Dj ::

51

Proof: Application of the procedure used in the proof
of Lemma 4. 2.1 (calculate n-l successive derivatives of the first

row of (4.1.1) eliminating x ,x , . . . ,xn) results in

2 3

Xd : X + 7.(xl, t) (4.2.4)
where (B of Lemma 4 2 l is the unit matrix) X' - [x X”) X(n-D]
. ll . . . d — 1,1 ,..., 1 ,
' ' <1) (1)
X : [xl.x2. - - . .an. Z (X1. t) = [0.fl(xl) + ql(t).f1(x1)+fz(xl)+ql(t)+q2(t).- - - .

n-2 n-2
f(l )(x1) + ...+fn_l(x1)+q(l )(t)+...+qn_l(t)]. The theorem

“WW// it" saw

follows by substituting (4. 2.4) into (4. 1.1).

Lemma 4. 2. l is one of the important features of this
section. The lemma is in a sense, an existence theorem. That is,
l in the system (4.1.1) satisfy the

condition det (B) J 0, then there exists a nonsingular transformation

if the aij entries of matrix A

(4. 2. 3). Identical conditions on the aij entries have been found

for the transformation to exist, when the normal system is linear.

Theorem 4. 2. 3 shows that the transformation (4. 3. 3)
converts the normal system (4.1.1) into an n-order reduced

differential equation .

Theorem 4. 2. 3: Consider the system (4.1.1) with p : n,

 

det (B) 7! 0, where matrix B is defined in the hypothesis of Lemma 2. 2. 3.

Then there exists an n—order reduced differential equation
dn 2:1 dn‘J n‘l n-l
—-—n xr1 : .2, a. 11-J xn + Z Pj(D)fln_J(xn) + E P.(D)qln_.(t)
dt 1:1 J dt j:0 j:.-0 J J

 

(4.2. 5)

52

-2 ,_
where 111(11n ) ._ f1(11111)’1l(11) A21A11F12(xn)1_ 2,3,... ,n,

1-2

C1,,(t) — qn (t) q,, (t) - AZlAllQl(t)1: 2,3, . . . ,n, corresponding

to the system (4.1.1).

Proof: Substituting the transformation (4. 2. 3) into the

system (4.1.1) results in

- -1 -1 , -
Xd ‘ B11’1‘1B X1(1'13’11’11‘1B 211%") 1 13111“an 1 B110“) +26%”)

(4.2.6)

, , 1 (n-l)
whereX :[xn,X1d]:[xn,x:1),...,xn ].

d
. (1) <1)
[8,, N<x > + B,,Q<t> + 2(xn.t)]' =fx,[,,( )+q,,<t>. ,,(x) +f ,,<xn>+q,,(t)+q,2<t),

(11111“) (11- -1)(

lnn(x ) 111111 t)+...+qln(t)], Zl'(xn,t) -:

° )

[f,,<xn) + q,,(r>. f‘,1J,<:n) + f,2<x 1+ q[,’(t> + q,,<t.) ..

f(1111-2) . ' (n— 2)
(x n)+ .+ f,n_,(.xn)+q,, (t)+ +q,,,_,(t)] and
> _ T ' _ .1-Z ' I -— _
111(11n) _ 111(11n)’ 111(Xn1— A211111111112(11n)1 — 2’ 1 1 1 ’n’q1111)" q11(1)’

ll

i-Z .
C111“) — A21All Ql(t) 1 2,3, . . . ,n. The theorem follows

since (4.2. 5) is given by the last row of (4. 2. 6).

The determination (Lemma 4. 2. 2) and application
(Thm. 4. 2. 3) of the nonsingular transformation Xd : B111 Z(xp,t) as
defined in the results of Lemma 4. 2.1 are unique to this thesis.
The technique employed in the proof of Thm. 4. 2. 3 offers not only

a new method of formulating the n-order differential equation, but

also of equal importance,a closed form relationship

53

-l
X .-:Bll

(4.1. 2) to the solution of the normal system (4.1. 1). This last

[Xd-Z(xn,t)] which joins the solution of the n-order equation

property is demonstrated by Thm. 4. 2. 4, which follows.

Theorem 4. 2. 42 If (4. 2. 5) is the reduced differential

 

equation corresponding to the system (4.1.1) and xp(t) is a solution

on I of (4.1. 2) then
[Xd(t) - Z(xp,t)] (4.2.7)

where Xe“) = [xp(t),x:)1)(t),... ,xp (0], z'(xp,t) =

(1) (1)
[0,fll(xp,t) + qll(t),fll(xp,t) +flz(xp,t)+qll(t) + q12(t),...,

fanl‘zhxpm) +. . .+ {In-1(Xp’1) + q(lnl112)(t)+...+ qln_l(t)] is a

solution of the system (4.1.1) on I.

Proof: By Definition 4. 2.1 the last row of (4. 2. 6) is

(4. 2. 5). By substituting (4. 2. 3) where X and Z(xp,t) are defined

d
after (4. 2.7), in (4. 2. 6), (4.1.1) results Since dezBXq3+ 21(xp,t),

l __ . ’1'
N<111’1)11[fll(xp1t)’flz(xp’1)’1 11’fln(1\ ,t)] (B11) and

P

Q'(t) : [qll(t)’q12(1)’1 . . ’qln(t)] (B11)' . Thus if xp(t) is a solution

of (4.2. 5) on 1, then X21“) = [xn(t),X'ld(t)] = [xn(t),x1(11),...,xfln-l)(t)]

is a solution of (4. 2. 6) on I, which implies that (4. 2.7) is a solution

Of (4.1.1) on I.

4. 3 A Relationship Between the Parameters and a Solution

"-

 

of a Class of N-Order Nonlinear Differential Equations

 

Nonlinear differential equations of the form (4. l. 2),

where T(xp) is a polynomial, are considered in what follows.

54

A relationship between the parameters and a solution in the form of

B0 + Blsin(;ot + ¢) of this class of differential equations is obtained.
The parameter-solution relationships determined in
Thm. 4. 3.1 are referred to as design equations when applied in
Chapter V to the design of tunnel-diode amplifiers and oscillators.
Before considering the details of the theorem, reference should be
made to Thm. C. l in the appendix. In Thm.C. l the nonlinear pa rt
Pl’n-1(D) f(y) of (4. 3.1) is derived for f(y), a polynomial of order

m.

Theorem 4. 3. 1: If y(t) : BO +Blsin(;ut + (0), BO :/ O, is

 

a solution on I: It) >tO of

 

 

 

n n dn-i
V 2.3a. _ y+P _ (D)f(y) +F(t) (4.3.1)
dtn 1:1 1 dt11 1 l,n l
- J
. . _ n-1 n-2 n-3 J_ d _
\xherc pl'n-1(D) — D 11dlD -dZD -...-dn_l,D _?,P1,0(D)—l
m .
and f(y) : z (1.39, a $0,170): q + q sin(;st + 9) where s>0
ij J m o 1
then
(1) qo : 1an Bo + dn-l bo
n .
(2) q cosG : B uncos(§l+ ¢) -B '> a .6111 “(1111)“ + (D) -L b
l l 2 11:11 1 2 l l
n .
(31 q sine = B wnSiI1(1—1—E—+ (a) — B v a an’lsihU—“fll + 91) —s b
l l 2 11:11 i 2 l l
(4) bS:O,S;£O,l s=2jor2j-1.
Where b and b . are defined in the results of Lemma C.2 and

2j 23-1
Ll’ Sl are defined in the results of Thm. C. l.

55

 

Proof: Since y(t):BO + Bl sin(wt + (6) is a solution of
dn n dn-i
(4.3.1) on I. [11 >10 , :11? (130+Blsih(st+¢)) : 1131111dtn’1 (Essen/11+?!» +

pl,n-1(D) f(BO+B151n(wt+¢)) + q0 + q181n(wt+9).
Calculating the indicated derivatives, and applying the

conclusion of Thm. CL this equation reduces to

n

 

In . r31 _ V 'n-i . (n-i)1r _ _ . ‘ .
[Blu Sin( 2 +¢) ijlaiBlsg s1n( Z + ¢) Slbl q181118]c0s tot +
[B wncos(-1-1-T—r+ ¢) - ‘1“1 a B '..111-1cos((241—111-T + Q5) - L b - c056] sin 't +
1 2 1:11 1 1 2 1 1 q1 “’
k k
n-lbo - anBO-qO 423 Mijj cos 2):..1t -.Z‘, ij2j sin ijt -
171 1:1
r r
2 Lb. sin 21-ltt-E S.b. cos 21-l't20.
1:2 121-1 (J )111'=2 121-1 (J b

Grouping the coefficients of cos wt, sinwt, cos ijt, sin ijt, cos(Zj—l);ot,
sin(2j-l)wt and constants, and equating each group to zero results in

(l), (2), and (3) conclusions of the theorem and the following

relations:

M.b.:O N.b.:0 ‘:l,2,...,k
121 .121 J

:0 Sb

L1 . . _ :0
1121-1 J 21-1

j:2,3,...,r.

Since Mj(2jt.1) is a polynomial in 231;.) of order n-l , n_> l, with
coefficients not all zero, M.(2j;.1)bZj : Oj : 1,2, . . . ,k for :1) >0,

J
. : O for J : 1,2, . . . ,k. The theorem follows by

implie S that bZJ

aPplying the same argument. to Lj((2j-l)u1)b2j_l.

56

 

 

 

Corollary 4.3.1: pr - qf 0, 2,4,. . . ,B1£0 then
(1) Boan : .q0 + dn-lbo
—[Wlsir1((—'------11-2q)Tr + 9)) " W2 cos((niqh + m]
(2) a :
p wn-p Sin (p-czmr

 

[wlsin((n'p)1T + (D) - W cos (Ln-11)-TL + (0)]

 

 

2 2 2
(3) a 1 n- I - hr
q w p sin 1.9—.—
2
q . b
_ l n n-rr v ’n—i (n-i)1r l
where, W1 ——B—c058 - w cos(—5+ ¢) + .., ai.» cos( 2 +¢)+L1T3——
l s l
q . b
_ l . n . nn V n-1 . (II-1hr 1
W2 — -B-—l- 31110 - to 51n(—Z—+¢) + 1; aim 5111(-——Z——— +93) + 51 33—1
1‘, is the sum over all i f p, q.
s
Proof: This is a direct consequence of Thms. 4. 3.1

and hence the proof is not included.

Consider as an example of Cor. 4. 3.1 with n : 2

 

(l) B0212 : —q0 +dlb0
”b1 q1
(Z) 3133-]: + {9B1 Sln(¢ - 9)
(3)a r-w2+d El-:1-—1-cos(6-(l))
2 1 B1 B1

and from the results of Lemma C. 2 with m : 3 ( the derivation

following LEITIIDB. C . 2)

B2
_ 2 1 2 3 2
b0_aO+BOal+02(BO+—2—)+Boa3(Bo+-2-Bl)
2 3 2
b1- B1[al+ 2012130 + .13(3B0 +7,— 131 ) ].

57
The importance of the parameter—solution relationships
(I), (2) and (3) in Thm. 4. 3.1 will be demonstrated when they are
applied to arrive at design equations , in the design of tunnel—diode

amplifiers and oscillators .

A notable feature of the n-order differential equation
(4.3.1) s at the. nonlinear part f(y) is a polynomial of order m.

This r-_ suits in b0 and b in equation (I), (2), (3) and (4) of

l,

Thm. 4.3.1, being nonlinear in the specified B0 and B11 When the

pol 'nomial f(y) is of order m = 3 then b0 and b are given in the

1
example following Cor. 4. 3.1.

131. ‘ ’

V. 'ON DESIGN METHODS AND EXAMPLES

 

5. 1 Introduction

 

Design methods, which require the construction of a
normal system of differential equations having a specified solution,
are presented here. These methods are based on a portion of the

mathematical properties developed in the preceding sections of

this thesis.

To exhibit that the design methods apply equally well
to a very large class of physical systems, the methods are applied
to a normal system of equations whose parameters have been divorced

from those of a particular physical system.

The approach to the design is to View the system perfor-
mance as one specified component xi(t) of the vector solution, X(t),
of the normal system. Then, by mathematical techniques developed
in the preceding sections, the mathematical relationships which
must be satisfied between the parameters in the normal system and

the specified solution xi(t) are determined.

5. 2 Oscillator Design

 

Methods of oscillator design in terms of complex frequency

n (12’13). From

domain equationdLaplace Transforms) are well-know
Chapter II, the development of Theorems 2.3. 2, 2. 3. 3 and Cor. 2. 3.1

(Eq. 2. 3. 8) provides some new tools which can be applied to oscillator

design in the time domain.
58

 

 

 

59

The linear oscillator is assumed to take the mathe-
matical form of a normal system of linear homogeneous differential
equations. If y(t) is to represent the oscillator response then y(t)
must be a non—constant solution of the normal system for all time
t, t > 0. In addition, there must exist a constant? > O and time

tO > 0 such that l-

[y(t + nto) — W + (n+11t01l < 6 (5.2.1)

forallt>Tandn:O,1,2,... 37.

‘3,

The number tO in (5. 2.1) is called the period of
oscillation. The smallest values of T, To, for which (5.2.1)
is satisfied, is the rise time of the oscillator. The amplitude

of oscillation is lim Max y(t)
n-u-oo nt0< t < (n+1)tO

Two methods of designing an oscillator which has a
specified frequency of oscillation, amplitude of oscillation and rise
time are presented. Method 5.2.1 illustrates the results obtained
in Theorems 2. 3. 2 and 2. 3. 3 whereas, Method 5. 2. 2 applies the

results of Cor. 2. 3.1(Eq. 2. 3. 8).

Method 5.2.1: Oscillator Design
k X .t
(a) Construct y(t) _—_— 23 Pm _l(t) e 1 , tZO, from the
1:1 i

specifications.
(b) Apply Thm. 2.3.3 to y(t) of (a) and thereby

determine the coefficients 1131' j : 1,2, . . . ,n of

n n n-j
'51—'17)" ) 1: E Elm—E171??— (5.2.2)
dt j:l Jdt J

in terms of y(t).

 

60

(c) Relate the aj j : 1,2, . . . ,n entries to the aij

entries in matrix A of the system
X : AX (5.2.3)

where X' : [x1,x ,x ] , A - [ai

21 . . . n _ .] by applying

J
Thm. 2'. 3. 2.

(d) Relate the initial condition y(J)(t ) : C. ,
o J+l
j : 0,1, . . . ,n-l to the initial condition on the system

(5. 2. 3) by

X::Cf xd (5.2nn

,x ] , Xé:[xn,x1(11), . . . ,xg11l)] and

) '-—
whereX .- [xl,x2,... n

matrix G is defined in the results of Lemma 2. 2. 3.

Method 5. 2. 2: Oscillator Design
(a) Construct y(t) as in (a) Method 5.2.1.
(b) Form the matrix Fm frmn a fundamental set of y(t),

i.e. if y1,yZ. . .yn is a fundamental set of (5.2.2) then

r .
,(11 ,(11 ,(11
F = 31 32 "' yn (5.2.3
m . . . . . . .

- -l n—l

- .J

 

 

(c) Determine the coefficient matrix A of the normal system

(5. 2. 3) as

-l

‘1 1‘ o. u;2.m
m

AzG F
n

1

where matrix G is defined in the results of Lemma 2.2. 3.

61

((1) Determine the initial conditions for the normal system

as in (d) of Method 5.2.1.

The parameter-solution relationships obtained by either 1

of these two methods when associated with the parameters in

J. l—-—' .-- '
1

systems of differential equations corresponding to known network 1

configurations are referred to as design equations.

L. viz-J" '-

The basic steps of the proposed design methods are
illustrated in the examples that are itemized to correspond to the

design method being applied.

Example 5. 2. 12 (Method 5. 2.1) Specifications require

 

an oscillator with a frequency of oscillation f : 111/217 , aplitude of
oscillation c and a "small" rise time.

(a) A possible y(t) is

1111 .th X3t
y(t) : cle + cze + C36 (5.2.7)
Ce‘jw ce11j¢ . .
where C1 = T1 c2 :———2——-—, 111: + Jw,)\2 : -J;o and X3 : 413, where

:11, c, ¢,o.3 are real and positive anda3 is as large as possible.
(b) The coefficients aj of (5. 2. 2) for n : 3 as given

in the results of Thm.. 2. 3. 3

2
(->\ X +)\ X 112131-1111

azlfll‘flz‘“ 1213

1 2 3 3 E1‘2:
(5.2.8)

62

(c) Proceeding as indicated in (c) of Method 5. 2.1
the condition, det (B) i 0, in the hypothesis of Thm. 2. 3. 2 must

be satisfied. For this condition

d“ (B) : 331(a33a32+a31a121’332322) ”a32(a33a31+a31a11+a32a21)’1 O

the third-order equation (5. 2. 2), n = 3, is associated with the normal

 

system (5. Z. 3) and hence the a, entries (5. 2. 8) are related to the
J
aij entries of matrix A in (5. 2. 3) by I
+ z - : ,‘.
811111 8L22 ‘333 C13 a1 ,
( a -a a +a a a a +a a a a )- "2-
' 8‘11 22 2112 1133 3113 2233 32 2:3 ““"" ’ a2 12‘
2 (5. 2.9) ..
det (A) :- 413(1) ‘—‘ a3 ’
(d) In addition, for det (B) if 0, the initial conditions
as specified by (5. 2. 4) are -l
1 1 1 0 0 1 1 .
., .1» 1
x2 I 8‘31 332 a33 ’1 1
+ + + + ..
LX3 8L331131 a131‘8‘11 a32321 al338132 a31811211328122 2135;131:113 a325‘23J Ly

 

 

 

 

 

are in symbolic form —1
X(t ) = G x (t ). (5.2.10)
0 d 0

Any physical system which has a mathematical model in
the normal form (5. 2. 3), with parameters satisfying the design
equations (5. 2. 9) and (5. 2.. 10), will have a Specified response y(t);
where normal system x3(t) = y(t) specified. Note, when y(t) is given
the solution of the normal system is implied by (5. 2.10). The para-
meter-solution relationships (5. 2.9) and (5. 2.10) are further illustrated

when applied in the design of a "Colpitts Oscillator".

63

1A normal system of equations corresonding to the

Colpitts Oscillator

 

 

 

j: -
1 -:'
1111"

 

-—-+--———- - — ... ~~

..

Figure 5.2.1

where gm, (1 and rp are well-known tube parameters is

 

 

 

 

 

 

1 "‘ '1 - -1
11v o 0 Cl v,
Cl 1 L1
£1— v — - gm '1 .1..— V
dt C2 C2 Czrp C2 C2 (5.2.11)
1 .1. Li :3. 1
g L _J L L L L _ L- L _
Specifying that det (B) = Lit—12 i 0. Method 5. 2.1 applies and
rpCZL

the 11ij entries in the coeffieient matrix (5. 2.11) are related to the

specifications (design equations) by (5. 2. 9) as

 

 

 

l + B- a.
C r L 3
2 p
1 1 l R _ 2
LC 1' LC 1’ LC“ ‘ T ‘ .3 (5‘2‘121
1 2 2 p
gm + 1 .. ((1 +1) (I f
(:1ch clchr cl'CZLl-p 3

64

and the system (5. 2. 11) initial conditions are related to the

specifications by (5.2.10) as

X(to) : o‘lxdno)

,(21

whereX':[VC,V ,I , L

1 C2
determined from (5.2.10).

1 _ (1)
131’ Xd ‘ [11:1

] and the matrix G is

Note, in the normal system (5. 2.11) lL(t) : y(t), any
other component of the system model could have been specified
similarly by starting with the desired component in the last row

(provided det (B) i 0).

Example 5. 2. 2: (Method 5. 2. 2) Specifications require

 

an oscillator with a response as given by (a) of Ex. 5.2.1 where

c3 :: O.
(a) y(t) is given by (a), in Ex. 5.2.1, with c3 : O.
(b) The matrix Fm as specified by Method 5. 2. 2, is
’1‘1t 1.213
e e
_, (5.2.13)
Frn — X t X t
X e l x e Z
L l 2

 

 

(c) The matrix A of the normal system is determined
from (5. 2.6). First, the conditions on the aij entries
of matrix A in system (5. 2. 3) for matrix C to exist are
stated, i.e. det (B) 74 0. In the case of system (5. 2. 3)

where n 2: 2, det (B) = a f 0 is required. Second,

21

formulate the matrix C as in proof of Lemma 2. 2. 3 as

-.“..‘.‘| H."—

 

 

G

11

21

 

 

 

65

21

 

f—azzflt 1+>t 2

-)\,\—a

l

2

. , . -1° —1 .‘
Finally the matrix product G FmFm G is

(d) The specified initial conditions are derived from

(5. 2. 4) as

 

Hence, the final mathematical model for the oscillator, as

fi

111(10)

 

 

 

 

[1

 

.—

determined from the specification of y(t), is

..-

:1
dt

 

where aZl

Note, the specified y(t) corresponds to x

1

 

_

 

‘21

a

22

21

‘1...) *3

a21

a22

2

22 3

 

FI-

 

)—

Z(

X

l

y(gQ

(mo)

 

_1

 

c—i

(5.2.14)

 

(5.2.15)

(5.2.16)

(5. 2.17)

i O, and the initial conditions are given in (5.2. 16).

t) in the normal system.

The parameter—solution relationships (5. 2.16) and (5. 2.17) are

applied in the design of a "negative resistance

ll

oscillator.

 

of equations

 

 

 

 

1

 

 

 

66

 

 

 

 

Consider the following network and corresponding System

 

 

(5. 2.18)

To impose the mathematical restrictions (5. 2.16), (5. 2.17), on the

parameters of the physical system requires; first l/L a! 0 then

and finally

 

L

 

IL(tO)

 

.1

 

Note in this case IL(t) : y(t).

Example 5. 2. 3:

specifications on y(t) are given by (5. 2. 7).

2R2

w) 1(1)

~I7'IT—1

y(to)

 

 

 

9n )
1_ O -

(5. 2.19)

(5.2.20)

(Method 5. 2. 2) Suppose the

Fm (5. 2. 5) determined from y(t) is

 

X t

 

The fundamental matrix

(5. 2. 21)

67

For det (B) at 0 as specified in (c) of Example 5. 2.1 the calculation

 

 

A : o'lfi‘ F11G (5.2.6) is

I‘D 1T1

F'
o 1 o 11

-1
A e— o 0 0 1 o

. .. +

”1‘11 21‘3 (X 1X Z+1. 11‘ 3+X 21‘ 3) >11 )1 2+7) 3

L 2

(5.2.22)

where the Inatrix G is given in (5. 2. 10). Design equations for the
Colpitts Oscillator are obtained in this case by letting X l:+J';11,

X 2 : -j:.1, X 3 : -o.3 and relating matrix A (5.2. 22) to the coefficient
matrix of (5. 2. 11). The initial conditions are related to the Colpitts
Oscillator system of equations the same as in Ex. 5. 2.1. Note the

companion-matrix form of F F11 in (5. 2. 22).
m m

5. 3 Amplifier Design

 

Methods of amplifier design in terms of complex
frequency-dmnain equations (Laplace Transforms) are well-known
(12,13) . , ‘ . ,

. To illustrate the results of Theorems 3. 2. 2 and 3. 3.1
developed in Chapter III, two methods of amplifier design in the
tilne-domain are given.

The mathematical model of the amplifier is assumed
to take the form of a normal system of linear nonhomogeneous

differential equations (5. 3. 4). The nonhomogeneous part of system,

(rector Q(t), is assumed to contain a component of the form qi(t):qisin(1...t+0)

.. > O, which can be associated with the amplifier input.

68
If y(t) is to represent the amplifier response then y(t) is a nonconstant
solutio'l of the normal system for all time t, t > 0. In addition there

is a constant 6 > 0 and time tO > 0 such that

ly<t +1.10)- y(t +(n+11t01 l < 6 (5.3.1)

for allt>T and n: 0,1,2,...

The number tO is called the period of the amplifier
response and the smallest value of T, To which satisfies (5. 3.1)
is the transient time of the amplifier. The gain Gm of the amplifier
is the usual steady-state peak output divided by peak input, denoted
Inathematic ally by

lvlax y(t)

C1m : Nlax qi(t) (5' 3' 2')

for t > T. The amplifier bandwidth denoted by b will be

(all-:1?) _<__ b. If y(t) : yt(t) + yS(t), and yt(t) satisfies the homogeneous
part of an n-order differential equation (5. 3. 3) where r : n then yt(t)
is called the transient response of the amplifier.

Two methods of designing an amplifier which has a
specified response y(t) are presented. Method 5. 3. 1 illustrates the
results of Thm. 3. 2. 2, whereas Method 5. 3. 2 applies the results
of Thm. 3. 3.1. The first part of Methods 5. 3.1 and 5. 3. 2 have been

given previously in Methods 5. 2.1 and 5. 2. 2.

1N_/I_ethod 5. 3.1: Amplifier Design

 

(a) Construct y(t) = yt(t) + yS(t), t > 0, from the specifications,
k 1.- t—
1
where y (t) : E P (t) e , t > 0, is the transient
t 1:1 rni-l —

response of the amplifier.

69

(b) Apply (b) and (c) of Method 5. 2. 1, to relate yt(t) to the

aij entries in matrix A of the homogeneous system (5. 2. 3).

(c) Determine the nonhomogeneous part F(t) of the n-order

(d)

 

 

equation
(1n n n-i
n : Z: a. “_1 y +F(t) (5.3.3)
dt J12]. J (it

by a theorem frozn Sec. B. 2.

Formulate the nonhomogeneous part, vector Q(t), of

the normal system

)1( : AX+Q(t) (5.3.4)

where X1 1' [X1,X2,o .. 1X11] , Q1(t) : [ql(t)9q2(t)9"' yqn(t)]:

A : [aij]’ by letting

Q‘(t) : [f1(t), ..., 111(1)] (of1 L’11' (5.3. 5)

as determined in Thm. 3. 2. 2. Matrices G and L
are formulated in the proof of Lemma 3. 2. 2. Parameters
fl(t), f2(t), ...,fn(t) are the components of the nonhomo-

geneous part F(t), (5. 3. 3) and satisfy Def. 3. 2.1, Eq.(3. 2.3).

Relate the initial conditions y(J)(tO), j : 0,1, . . . ,n-l
to the initial conditions on the normal system (5. 3.4) by

x : o‘lxd (5.3.6)

which is defined in the results of Lemma 2. 2. 3.

70

Method 5. 3. 2: Amplifier Design

 

(a)

(b)

(C)

((1)

Construct y(t) as in (a) Method 5. 3.1.

Form the matrix Fm from a fundamental set of (a)

as in (b), Method 5. 2. 2.

Determine the nonhomogeneous part F(t) of the n-order

equation (5.3. 3) as in (c) Method 5.3.1.

Determine the matrix A and vector Q(t) in the normal

system (5. 3. 4) as

A .—. (3’11? F11 o

m 111
d -1 (5.3.7)
om Fm 3mm X(t) 1

where notation is defined in (3. 3.4), (3. 3. 5) and (3. 3. 6).

The initial condition on the normal system will be the

same as in (e) Method 5.3.1.

The parameter-solution relationships obtained by either

of these two methods when associated with the parameters in systems

of differential equations that correspond to known networks con-

figurations, are referred to as design equations.

The basic steps in design Method 5. 3.1 are illustrated

by an example that is itemized in correspondence to the design

method.

Specifications require an amplifier with a transient

response yt(t) : cle-atsinmt + (D) and gain Gm for a frequency range

.<...
u) “0&2

1___

71
(a) A possible y(t) is

y(t) : cle-a1+c2e-atsin(u1t + ¢) + Blsinwt (5, 3.8)

where c1,cz,a.,:.1, and B1 are real non—zero constants.
(b) First the coefficients a). J : l, 2,3 of (5. 3. 3) are
related to yt(t) as in (b) Method 5. 2.1. Next specifying det (B) i 0

(as in (c) Method 5. 2.1) the a]. coefficient are related to the aij

1
entries in matrix A of (5.3.4) as :
a11132211333:“3“1‘1511 j
-(a a -a. 21 +21 a -a a +2.. "1 -a a ): -(3a2+w2) :a
1122 2112 1133 3113 2233 32 23 2
det (A) : — a(o.2+u12) : a3, (51319)

(c) Proceeding as indicated by Method 5. 3.1, a possible
F(t) for the third—order equation (5. 3. 3) is found in Thm. B. 2. 3
where In : l and n = 3 as

F(t) 2: ql sin(;..1t + 8) (5. 3.10)

 

2' 30:13

0. —2w

where ql : B (1)/(302.1); + (2:11 -o.111) , 8 : tan-1

1

(d) The nonhomogeneous part, Q(t), of the normal system

(5. 3.4) is related to the nonhomogeneous part, F(t), of (5. 3. 3),

n : 3, by
-l -l
Q'(t) : [fl(t),f (t),f3(t)] (G L )' (5.3.11)
k-2
where 23 P1.1(D) f .(t ) : 0, t on I, k : 2,3,
j:0 J k—l-J 0 O

as specified in Def. 3. 2.1. This specification requires

K1
[\1

that the components of F(t) satisfy

(t )= 0

2
pom) f1110) ‘1' fl 0

3 3
pom) 12(10) + 121(1)) fl(t0) : (D + £3Z)fl(to) : 0

Let fl(t) : fa(t) : 0, f3(t) : qlsin(a. t+ 6). The matrix G is specified
by (5. 2.10) and matrix L.1 : [flij] is determined,as in the proof of

Lemma 3. 2. 2,as

 

 

r l o 03
L'1 ~ a l 0 (5 312)
_ 33 . .
az+a a +a a a 1
_ 33 31 13 32 23 33 J

The vector Q(t) (5. 3.11) can now be calculated as

 

 

r r _ 3' . ‘1
'1 a3zq1 s1n(...t+ 9)
ql(t) d
a31q1 sin(<‘«t+ 9)
Q(t) : q2(t) : d (5.2.13)
q3(t) 0

 

 

 

 

where d : a3l(a3111112+a32a22)1 1132(113111111 11321121)‘
(e) Finally the specified initial conditions y(J)(tO) : Cj+l

.1 = 0,1,2, are related to the normal system by X(to) : G111

Xd(t0) , as
implied by (5. 2.10).
Any physical device having a mathematical model in

the normal form (5. 3. 4), with parameter-solution relationships

as specified in (5. 3. 9), (5. 3.13) and (5. 2.10) will contain the

73

specified function y(t) as a component of its vector solution, in

this case x3(t) : y(t). These parameter-solution relationships
(design equations) are applied to the design of an amplifier.

The normal system (5. 3.14) corresponds to the network

shown in Figures 5. 3.1,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

F ‘ " 1 —1 " ” ‘ ' “
V1 0 C_1 C—l V1 0
d -1 -(rp+R4) 0 "H e (t)
I = — I +
Elf 4 L4 L4 4 L4
—R
l 3
1 _._ O _— I O
1.. 3 ... _L3 L3 .1 ... 3 .1 b _1
(5.3.14)
R4 L4
MM +~
1 L3.
+f111 Cl __ o
R
e 3
g
1 v
Figure 5.3.1
If det(B) : ——21—— i 0, Method 5. 3.1 applies, and the
L3Cl

relationship between the entries in (5.3.14) and apecifications
(design equations) (5. 3. 9), (5. 3.13) and (5. 2.10) are

(rp+R4) 3

+——— :31
L4 1.

 

 

74

 

 

R
l 1 1 3 2 2
——(——— + ——-) + (r +R )2 30 + 61 (5. 3-15)
C1L4 L3 L3L4 4
-(R +r +R)
3 p 4 — (1(c1 +1.12)
L31114Cl

Ll}. — ‘ ,
L4 eg(t) _ L3Clql Sln(61t+ 9).

The initial conditions for the system (5. 3. 3) are
'1 , (1) (3) 1
G [y(to). Y (to). y (to) ].
The gain and frequency (bandwidth) are related by the

ratio of y(t) and eg(t), from (5. 3.15), as

‘1‘ . (5.3.16)

G : - _
1n 2 . 2 2
L3L4C1a V(3aw) + (263 - c1 )

Note, in the system (5. 3.14) where 13(t) is the current

through L3, 13(t) : y(t). Any other solution component of the normal
8 ystem could have been specified similarly by starting with the

desired component in the last row.

Nonlinear Amplifier and Oscillator Design

The mathematical properties developed in Thm. 4. 3.1

and the process specified in Def. 4. 2.1 for reducing the normal

Sys tem (4.1.1) to an n-order differential equation of the form (4.1. 2)

a 1’8 used in forming a method of designing nonlinear amplifiers and

oscillators in the tilne-domain.

75

The mathematical models of the nonlinear amplifiers
and oscillators are characterized by a discussion similar to that
afforded the linear amplifier if the word "linear" is replaced by
"nonlinear". In addition, it is assumed tint the nonlinear oscillator
contains only constant entries in the nonhomOgeneous part of its
niathelnatical model.
Suppose an amplifier (or oscillator) is to be designed with
a specified response y(t). A method of obtaining a normal system
of nonlinear differential equations which has a solution satisfying
the specification is:
(a) Construct y(t) from the specifications.
(b) Apply Thm. 4. 3.1 to relate y(t) to (I) the
coefficient a), j = l, 2, . . . , n, (2) nonlinear part
f(y), and (3) the nonhomogeneous part F(t) of the
n-order equation (4. 3.1).
(c) Construct the reduced differential equation from
the normal system (4.1.1) by the method specified
in Def. 4. 2. l. Equate the coefficients and
parameters determined in (b) to corresponding
parameters aj j : l, 2, . . . , n-l, F(t) and T(xn)
of the reduced differential equation.
(d) Relate the specified initial condition y(j)(t0)
j = 0,1, . . . , n-l, to the initial condition on the
normal system (4. 1.1) by means of the nonsingular

tr ansfor mati on

-1
XzB11[Xd

as formulated in tre proof of Lemma 4. 2.1 and

- Z(xn, t)] (5.4.1)

applied in Thm.. 4. 2. 4.

76

This method is illustrated on a general system of
equations to obtain parameter-solution relationships which
are then applied in the design of a tunnel-diode amplifier and
oscillator.

Specifications require an amplifier with a gain Gm

for frequency range wl_< :11 < 1“”2 .
(a) A possible y(t) is

y(t) : BO + Blsin(63t + (D) (5.4. 2)

the same y(t) is suitable as an oscillator response where B1 is
the amplitude of oscillation and f : 6.1/211' the frequency of oscillation.

(b) The coefficients ai, i : 1,2, the parameters T(y)

and F(t) of . 2 .
d2 a dZ-l
“‘2 y = 2: 1 ..,—T y + T(y) + F(t) (5.4.3)
dt 1:1 dt

are related to y(t) by

 

-bl q
al:'—B— + NB 5111(¢-9)
l l
d b q
2 l 1 1 _‘
8.2—“:3.) ‘11—‘15.— -:—B— COS (9 -¢) (5.4.4)
1 l
m j
T(y) = Pl,n-1(D) f(y) = (D-dl) .3 ajy
1:0
2 d1b1 ql
F(t) : dlbO—BO[—61 1111-6—1— — B—icosw -¢)] + qls1n(tot + 6).

The notation will be found in Thm.4. 3.1 and Cor. 4. 3.1. Note, the
nonlinear part T(y) is a polynomial of order m. The b0 and b1, are
defined in the results of Lemma C. 2 and have the form of the b0 and

b1 in the discussion after Cor. 4. 3.1, namely, they are nonlinear in

the specified Bo and B11

«J

«J

(c) Construct the reduced nonlinear differential
equation corresponding to the normal system

(1 1‘ '1 1) '- ‘1 '- T
X1 a11 {le {11"2) (1119
+

d
Eff + (5.4.5)

 

 

 

 

 

 

 

 

X2 a21 f2("2) ‘12“)
L - L

as specified in Def. 4. 2.1.

First, specify the condition det (B) : aZl i 0, formulate

Xd : Bllx + Z(xn, t) as in the proof of Lemma 4. 2.1 for this

C2188

(5.4.6)

 

 

 

 

 

 

 

)1; a 0 x f2(xz) + (12(1)

Solve (5.4. 6) for the vector X. Substitute the vector X

(5.4. 6) into the system (5.4. 5) to obtain

 

 

 

 

 

 

(5.4.7)
px21 10 1 11 x23 1— 0 G
d
at 7' ‘1
LX2_ L0 a114 (_xz Lazlflb‘zl‘L‘D‘H11f21x21+a21q1m+pfilll(12(1),,

the last row of which is the reduced differential equation. Equating
the coefficients of the reduced equation to corresponding parts in
(5- 4.4) provides the desired parameter-solution relationships

(de sign equations).

Coefficients of matrix A12

 

.131 q

all—T3— +(B sin(¢-9) (5.4.8)
1 l

321: 2121"0

 

Coefficients of vector N(x2)2

 

b q b
2 .
fl(XZ)——-——[-a3 413—1- —’B1 Slli(¢-8))——l— -
21 l ... l l
ql
B” cos (0 - (0)] y
1
m 1
f (x ) _ 1) a y
2 2 :0

Coefficients of vector Q1(1)1

 

 

1 b1 ‘1 .
qltt) 2;; - (~B—l - ”Bl Slum - e) ) b0 ..
Bo[‘”1' (3113-1 - :31 sinw - e) ) :13}. -
l 1 1
25-1; Cos (0 -¢)] + ql sin («t + 98
qz(t) = 0.

Where b0 and bl’ as defined in the results of Lemma C. 2, are

nonlinear in the specified B0 and B11 It is important to note the

s ystem parameters (5. 4. 8) have been obtained for the case where

the nonlinearity is a polynomial of order m. The effect of the non-

l inearlity is apparent in b0 and b1 which contain the specified B0

. . m
rand B1 and powers of these constants up to and including B0 and

m

Bl

(d) Finally the initial conditions for the normal system

(5. 4. 5) as obtained from (5.4. 6) are

n1

1 . v '
x100) = 3: [y(to) - £0 aj 1”](t0)]
(5.4.9)
xz(t0) = y(to)

79

Any physical device having a mathematical system of
equations in the normal form (5. 4. 5), with parameter-solution
relationships (design equations) as specified in (5.4. 8) and (5.4. 9)
will contain the specified response y(t) as a component of its vector
5 olution (in this case x2(t) : y(t) ). The parameter-solution relation-
ships (design equations) (5. 4. 8) and (5. 4. 9) are applied in the following

to the design of a particular physical system.

A normal system corresponding to the tunnel-diode

11 etwork of Figure 5.4.1 is

 

 

 

 

 

 

 

 

 

 

 

 

 

1'— F _ .. " 7 " ' ”r “
: 1 1 _5 _._1. 1—1 r o ‘5“)
1 L L L L
l L
1
d I
_ 1 -
(it i — + +
1 1 O 1 , 11(11c) 0
i 1c C 1 1c C
L 11 7 ‘iL ‘1 F ‘ h "‘ (5.4.10)
1
__ - L
—l|—— C
f _-
(VC) T R

 

 

 

 

Figure 5. 4.1

The tunnel diode characteristic is assumed to be

3 2 ' . '2 '3
f(xC) —. (ng-hvo + IO) + (-g + 3h\O)\C - 3h\10\-C + hxc

(5.4.11)

“"hich is the idealized-tunnel-diode characteristic

8O

 

 

 

I

P

I

o

I

v

wlaere:
I +I v +\
I = p 11 x : 1
o 2 ’ o 2
_3 (Ip-IV) h- 2(IO-IV)
112‘ "'v‘” 11 (r-vHr—r)7
p 1v o 1v1p

Ip and v are the peak current and voltage, IV and vV

a re the valley current and voltage. The inter - relationship between

tlae entries in (5.4. 8), (5.4. 9), (5.4.10) and (5.4.11) are determined

 

 

 

 

 

 

 

 

 

as
_ G (6)
T1} :11: 1 wG ““111'91
m
(5.4.12)
_-_1_ __ C[_w2 _ (Gd1e) + sin(¢—O)) Gd(e) _ cos(e - (3)
L " C the.“ " c c.
m m
-f(\'c) : ‘3 a. \j
C £0 3 C
V (t) __ G (e) . _ 2
5L :c{_é( 21; +Slnigme) ) [(-g +7”:- hB1)e+he3+IO]-
G (e) - _ G (e) q
m

where: Gd(e) : g + 3heZ

f2w/21rande2B -V .
O O

81

3 2
+3}— hBl, Gain Gm r ql/Bl frequency

The initial conditions on the normal system

are then

H

C B o cos(mto+ m + f((VO'i' e) + B

ILUO) 1 sin(totO + G ))

 

 

l
VC(tO) : (v0 + e) + Bl sin (auto + Q5). (5.4.13) :
The oscillator design equations (ql :: 0) are:
23 _ Gd‘e’
L " C
2
G,(e)
Z
i —. c [a + “2 1
C
v C} (e)
g - d 3
_L. : Ci —E-z- [(-g+—ZlBl)e+he +1 ]+
Z
[Z Gd(e) \ (5.4.14)

It is important to note that three design equations
are presented in (5.4. 12) and (5.4.14) . The addition of the
equation for VS(t)/L in (5.4.12) and (5.4.14) is unique to this
thesis. In addition, the nonlinear function (a function of operating
point e , and a function of the specified Bl) Gd(e) is a generalization
of the results of Kim [8, p.416], who obtained by a different method
Gd(e) where e z: 0. The generality presented in this design technique
is apparent after examining parameter-solution relations (5. 4. 8).

The parameter-solution relationships (5.4. 8) as viewed
from the application point of view are a function of operating point and

a tunnel-diode characteristic which is approximated by a polynomial

of m-order.

VI. C ONCLUSION

 

The first parts of Chapters II and III develop the
mathematical properties which relate the solution of the normal
system of linear differential equations (1. 2.) to the solution of an
r-order (r i n) differential equation, (1, 3). The foundation of
this development is presented in the proofs d Theorems 2. 2.1 and l
3. 2.1. It is proved in these theorems, by applying a transforma-
tion of the form

...1 _
X=C [YS-L1H(t)] (6.1)

‘l’b. '-’- ———“’:-— - ; 'x
z: .' _ A.
l

4 |

. l -l
where X' = [Xl' x2, . . . , Xn] and Y5' = [0, 0, . . . , O, y, y( ),..., y(r ), 0, 0, ...,0]
to the normal system (1. 2), that there exists a set of s differential
equations, 1 E s _<_ n, of r-order associated with the system. These

results are extended in Theorems 2. Z. 3 and 3. Z. Z.

In Theorems 2. Z. 3 and 3. 2.. 2 conditions on the aij
entries of matrix A in the normal system (1. Z) are given so that
there exists a differential equation of n-order associated with the
system. In the proof of these results a technique for formulating
a transformation of the form (6.1), r = n, is given. Note, the
mathematical properties developed in these theorems allow the
determination of the solution of a normal system in terms of the

solution to the r-order (r < n) differential equation associated with

the system.

82

83

Additional mathematical relationships are developed
in the later parts of Chapters 11 and III to interrelate the parameters
in the normal system (1. 2.) to the solution of the r—order (r: n) dif—
ferential equation associated with the system. These relationships
provide the mathematical tools for relating the system parameters

to a component xn(t) of the system solution X(t).

The usefulness of the mathematical properties developed
in Chapters II and III are demonstrated in design methods
and examples of Chapter V. One method, which illustrates some
paramete r-solution relationships developed in Chapter II, has two
basic steps: (1) Construct (Thm. Z. 3. 3) an n-order homogeneous
differential equation which has a specified solution, xn(t). (2) Relate
the coefficients (Thm. Z. 3. Z) and the initial conditions, transformation
(6.1) where r:n and H(t) : 0, of the n-order differential equation to
the coefficients and initial conditions of a normal system of differential

equations .

In another design method of Chapter V some particularly
interesting results which were developed in Thm. 3. 3.1 of Chapter III
are applied. The coefficient matrix A and the vector Q(t) in the normal
system (1. 2), (3.1.11) are related to a solution xn(t) and the nonhomo-
geneous part F(t) of the n-order differential equation (3.1. 2) by an
expression of the form

A = G-l'E.‘ F ’10
m

m

 

84

This design method is completed by determining the initial condition
on the normal system by means of a transformation (3. 3. 6), of the

form (6.1).

Patterned after the linear case, it is proved in
Chapter IV that under certain conditions (hypothesis of Thin. 4. Z. 3)
a class of nonlinear differential equation (4.1.1) can be transformed
into an n-order "reduced" (Def. 4. 3.1) differential equation. A
solution of the reduced differential equation and a solution of the
corresponding nonlinear system are shown (in the proof of Thm. 4. 2.4)

to be related by

X : BllX + [.(xp, t)

(1) (n-l)

I 3 ' -— r I __

\xhcre xd _. [xp,xp ,...,xp ], x _ [xl,xz,...,xn], and B11 and
the vector Z(xp, t) are defined in the proof of Lemma 4. 2.1. These

results are applied in the design of tunnel-diode amplifiers and

oscillators in Chapter V.

The criteria det (B) i 0 as defined in the hypothesis of
Lemma 2. 2.. 3 was proved necessary for a transformation of the
form (6.1), rzn, to exist. This criteria is thus found in the
mathematical tools applied to the five design methods of Chapter V.
The development of other criteria for the existence of the transfor—
mation (6. 1) would be a useful extension of the results presented

here.

10.

11.

12.

13.

14.

15.

16.

17.

LIST OF REFERENCES

 

Brown, D.P. , Derivative explicit differential equations for RLC
graphs, Journal of Franklin Institute, Vol. 275, No. 6, 1963.

Wirth, J. L. , Time Domain Models of Physical Systems and
Existence of Solutions, A Thesis, Michigan State University, 1962.

Koenig, H.E. , Tokad, Y. , and Kesavan, H.K. , Analysis of l
I

Discrete Physical Systems, Class Notes for Pilot Program, E.E.
Dept. , Michigan State University. , 1962.. 1

Murray. F. , and Miller, K. , Existence Theorems, New York
University Press, 1954.

Moulton, F.R. , Differential Equations, Dover, 1958.

ware-M ~

Coddington, E. , and Levinson, N, Theory of Ordinary Differential
Equations, McGraw-Hill Company, Inc. , 1955.

Hurewicz, W. , Lectures on Ordinary Differential Equations,
John Wiley and Son, Inc. , 1958.

Kim, C.S. , and Brandli, A. , High frequency high-power operation
of tunnel diodes, LR. E. Trans. on Circuit Theory, Vol CT—8,
No. 4, 1961. .

Turnbull, H.W. , and Aitken, A.C. , An Introduction to the
Theory of Canonical Matrices, Dover, 1961.

Gantmacher, F.R., The Theory of Matrices, Vol. 1, Chelsea
Publishing Company, New York, N.Y., 1959.

Faddeeva, V. N., Computational Methods of Linear Algebra,
Dover , 1959.

Ryder, J.D. , Electronic Fundamentals and Application, 2nd ed. ,
Prentice—Hall, Inc. , 1960.

Valley, Jr. , G.E. , and H. Wallman (Editors), Vacuum Tube
Amplifiers, McGraw-Hill Book Company, Inc. , 1948.

Hohn, F.E. , Elementary Matrix Algebra, Macmillan Company, 1958.

Kaplan, W. , Advanced Calculus, Addison-Wesley Publishing
Company, Inc. , 1953.

Hildebrand, F.E. , Introduction to Numerical Analysis, McGraw-Hill
Book Company, Inc. , 1953.

Dwight, H. , Tables of Integrals and Other Mathematical Data,

Macmillan Company , 1961 .
85

APPENDIX A

 

THEOREMS AND DEFINITIONS FROM REFERENCES

Definition A. 1: The m n-dimensional vectors

j:1,2,...,1n

 

111
E
. i:1
Ci constant, implies Ci : O, i : 1, 2,. . . ,m,m <n.

are said to be "linearly independent" if the identity CiXi : 0, where

 

Definition A. 2'. A set of functions yj(t), j -_; 1, 2, . . . ,r, is said

to be "linearly independent" over the open interval 1 : [t2 tl<t<t2]
r

where t1 and t2 are constants, if the identity 3 C.y.(t) : O for all t
1:1

onIimplies Cj : 0, j : 1,2,...,r..

 

Definition A. 32 A set of functions yj(t) , j : 1, 2, . . . , r,
which are linearly independent solutions of the r-order equation

(2. 1. 2) on I, is called a ”fundamental set" of (2.1. 2).

Definition A.4: If Fm is a matrix whose 11 column are n

 

linearly independent solutions on I of (2. 1. l), the normal system

X : AX, then Fm is called a "fundamental matrix. "

Theorem A. 1: [6, p.70] If 11 is a fundamental matrix of (2.1.1)

 

the normal system X : AX and D a(comp1ex) constant nonsingular

matrix, then HD is again a fundamental matrix of (2.1.1). If H1

and H2 are fundamental matrices then H1 : HZDZ. where D2 is

nons ingula r .

86

-1- ..- u t..- 1..!“

?,.__-

87

Theorem A. 22 [7, p.47] The r-order homogeneous dif—

 

ferential equation (2. 1. 2) always has a fundamental set (Def. A. 3)
of precisely r solutions. A set of r solutions of (2. 1. 2) on I constitutes

a fundamental set if and only if its Wronskian

Y1 YZ yr ‘1
1 l 1
ya) YE) Y1.)

VWYI’ yZ, . . . , yr)”:

 

 

y(Ir-1) y(Zr-1) (r-l)

— —

is nonsingular (WfO) for some tO on I.

Theorem A. 31 [14, p. 215] The coefficients of X r in the

 

characteristic polynomial of matrix A, det [A-XI]is (-1)r times

the sum of the principal minors of order n—r of matrix A, where

A 7: [aij] is of n-order and I is the unit matrix. In particular, the
coefficient of X n is (-1)11 and the constant term in the characteristic

polynomial is det (A).

Leibnitz‘s Rule: [15, p. 219] Let the function f(x,t) be

 

continuous and have a continuous derivative in a domain of the x-t

plane which includes the rectangle a<x_<_b, tl< t< t2. Then for

t1<t<t2

"raw

clmulb

88

Theorem A.42 [15,p. 220] Let the function f(x, t)

 

satisfy Le ibnitz ' 5 rule .

In addition, let the functions a(t) and b(t)

be defined and have continuous derivatives for t < t < t .

 

1 2
Then
b(t) b(t)
Ei’lf f(x,t) dx : f[b(t),t] 533%) - f[a(t),t] (133),, 3% x,t)dx
a(t) a(t)

APPENDIX B

 

SOLUTIONS OF CERTAIN DIFFERENTIAL EQUATIONS

 

The design techniques developed in this thesis require
that a specified system performance characteristic be considered a
solution of an n-order differential equation. This necessitates a
knowledge of the inter-relationship of the solution of an n—order
differential equation to the coefficients of the n-order differential
equation.

The n-order homogeneous differential equation

 

 

y (B. 1)

is known [7, p. 65] to have the general solution

k X.t
y(t) = z Pm__1(t) e 1 (3.2)

where (a) Pm _l(t) is a polynomial in t of degree mi-l,

1
(b) x i’ i : 1,2, . . . ,k are the distinct zeros of the polynomial
n . k
L(,\) : Xn- 2; ai xn-l : 0 each of multiplicity mi and (c) )3 mi : n.

1 1:1

 

dn n n—i
—-—-—-y: E a. _. y+F(t) (B.3)
at“ 1:1 1 d1“ 1
is known [6, p. 87] to have the solution
Y(t) = yh(t) + yp(t) (B-4)

89

 

 

 

.. la '. w 7'“ '7 fire—“m

{way «43;» .

.
J .
iv '

()0

n Wk(s)
where (a) yh(t) satisfies (B 3),(b) yp(t) : 2) ¢i(t) W( ) F(s)ds
i::1
t
o

and (1) ¢i(t)’ i: 1,2, . . . ,n, is a fundamental set for(B. 3)when F(t) : 0.
(2) W(s) is the Wronskian 01(13. l)and (3) Wk(s) is the determinant
obtained from the Wronskian (2) by replacing the k—column by

(0,0,1).

When Xi in (B. 2) are distinct, i: 1, 2, . . . ,n, then
the particular solution y(t), y(to) : 0, t0 on I can be expressed by
n X .t -X .s

y(t) : 2 Aie 1 F(s)e (15 (13.5)
1:1

 

n+i
where Ai : (-1)
n 2 p >i >j > 0
B. 1 Inter-relationships of Coefficients of N—Order

 

Polynomial to Zeros of Polynomial

 

The design methods of Chapter V necessitate a knowledge
of the relationships between the coefficients and a solution of an n-order
homogeneous differential equation. This information is supplied in
Thm. 2. 3. 3 as a result of deveIOping the relationships between the

coefficients and zeros of an n-order polynomial.

..._- e. K ,. ‘6',
1‘”. '

91

0

Theorem B.l.1: If.\.

1
n . k
n .-

 

L()\) : X n+3 b.)\ J, of multiplicity mi, '>‘ mi : n, then
j:

1 J i=1

blz-(rl+r2+... +rn)

b2: (r1r2+...+ rlrn+rgr3 +...+r2rn+...+r

n-l

bn :2 (-1)n(r1r2 . . . r)

11

1

j
where ri = .\ ., i :

'fiM

1

Proof. By hypothesis

11

14K)::Xr1+ :3 batn'J
1:1 J
m m m
l 2 k
= X—X — -
( l) (1,12) ...(x xk)
j-1 1-1 J
Ifr.=)\.wherei: E m +1, E m +2,...,Sm andj=1,2,...,k
l J p=1 p p—1 p psi
then
L()x) — ()x -rl) . . (,\ -rm )(\-r +1). .(,\ “rm+1n)' .(X—rn)
1 l 2
—).n + + + )xn'l+( r + +r r +r r +
_ —(rl r2...r. r12... 1n 23...
+ + + )1” 1 W r )
r2rn n-lrn ' - 1.1 2" Iin

This implies the theorem.

1'
I]

.,kisazeroof

(3.1.1)

j-
1711-1, Z) 111 +2,..., Zm,j:1,2,...,k.
P p= P

l..- - ”Aux—0-x”.
I
. ‘. r"

a. .«u-l
‘ . .71”
.1!

Corollary B. 1.13 Ifxi, 1:1,2,... , n are distinct

 

 

n .
roots of L(>\) : kn + E b.)\ n-J then
1:1
bl:-()\l+)\2. .+)\n)
b2:+(xlx2+...+xlxn+xzx3+ +x2xn+ +xn_lxn)
(B. 1.2)
n
bn .. (—1) (1.112 kn).
Proof: This is a direct consequence of Thm. B.-1..1.
Theorem B. 1. 2: If hi, 1: 1,2, . . . ,r are distinct non
n .
zero, zeros of L()\) : Kn + :3 b.>\n-J, then r coefficients of L(>\),
1:1
b., b are explicitly related (B. 1.4) to the remaining

J j+1""’b_j+r-l

n—r coefficients.
Proof: By hypothesis

1']. .
L()\.)=)\I.1+ :3 1).).‘1'3
1 1 . J 1
I 3:1

:0 fOI‘l: 1,2,...,1‘.

This system is written in matrix form as

Va : L
',_ ,-_ n_ n-i _ n__v
wherea _ [bj,bj+1,...,bj+r_l], -[ x1 .Ssbixl tr :bixr

Sis the sum over all i, i #3, j + 1,.. . , j+r~1 and
s

n-i ],

 

[- n-j n-j-l n~j~r+l~
X1 '\1 X1
n—' n-'-l ..'..
V; )(23 x23 ngr“ (13.1.3)
Xn-j x n-j—l \n—j-r+l
_ r r ' r _

 

Forj : 1, r :2 n (B. 1.3) is the Vandermonde matrix [16, p. 85].

Forr<n

det (V) 2: k V1

where k I [Kill-J-r-H, Xn-J-r+l

n—j-r+1]
Z ,..

.,Xr , V (X.-)\j).

1:8:r:i>j 1

V1 is the Vandermonde determinant [14, p.47] which is non—zero

since Xi )4 \j Therefore

a: V L (B.1.4)

which implies the theorem.

B. 2 Solutions of N-Order Nonhoxnogeneous Differential

 

Equations

The relationships between the coefficients and a
solution of an n-order nonhomogeneous differential equations are
now determined. Formulas are given such that if a solution is known,
some or all of the coefficients of a corresponding differential equation
are specified. The relationships determined are for particular solutions
in the form of power series, linear combinations of exponential
functions and linear combinations of sine and cosine functions. The

results of this section are applied in the design methods of Chapter V.

q ... .... -—.F.‘Ll-‘ u~-fifirm
. . ~
. . .

94

 

 

 

 

 

 

m .
Theorem B.Z. 1: If y(t) : 2: BjtJ is a solution on I:
1:0
It) >t _. of
O
n n n-j
d . .
_._; : Z dn x+F(t) (B.2.1)
dt j:l J dt
IT] .
where F(t) : E q.t.J then
jso J a
(14m)I (fin-l)! , j! , )
qj T j+n —T]"____ j+n-ld1—H°-j~! Bjdn :-
(13.2.2) 1
wherej20,1,...,1n,Bk:0,k>m. .j
m
Proof: By hypothesis y(t) : E B tJ is a solution of
1:0
(B. 2.1)0111:' )t) >t0, therefore
n m . n n-1 m . .
dn (:Bch): 2 ai dn_i(._,B.tJ)+ eqJ
dt j:0 J 1:1 dt j:0 J ij J
dr 1T1 - In ._r
Since r (z: B.tJ) —_ :3 (j) 0-1)... (j-r+l) tJ .
dt er J Jso
m '-n m '-n+i
:: B.(j)(j-l)(j-Z)...(j—n+l)tJ : Eai :3 B.(j)(j-1)...(j-n+i+1)t‘]
J's—.0 J isljso J
m . IT] .
+ a E B.tJ + 13 q.t‘].
“ 1:0 J 1:0 J

Grouping the coefficients of like powers of t and equating each to

zero the theorem follows.

Corollary B. 2.12 If Bin of hypothesis of Thm. B..2.1 is

 

not zero then

95

(1) a . ,an are explicitly expressed in

a ..
n-m’ n-m+1'

terms of q0,ql,. . . ,qm, for m<n-1

(2) a ,a ,an are explicitly expressed in terms of

1 2,...

q0.ql, .. . ,qn], for m : n+1

(3) ayaz, . . . ,an are explicitly expressed in terms of
qm_n+1~ qm—n+2’ . . . ,qm, for m >n-1.
Proof: The system of equations (B. 2. 2) in matrix form
is
Ma : q
, 3 1_ A. I: _ .. ..
uherea _[an-m’ n-m+1"°°’dnJ’ q [qo,..., qm] for m<n 1,
1_ ' t- _ ._ : ..
a — [al,az,...,an), q - [ q0,..., qm] for m n 1, and

a' : [al,a2,.. . ,an], q' : [_qm-n+1"" ,—qm] for m >n-1.

The coefficient matrix M is upper triangular in each case and has
(m! B )n
m

a nonzero determinent equal to Tm-11+1)'7n1-11+2)‘ (m), for m : n,

 

+1
(ml B )n (m! )m
m form>n and m form<n-1.
0.’ ll 2! m.‘ — U! l.‘ ml

 

 

This itnplies the corollary.

 

m [.1 .t
Theorem B. 2. 22 If y(t) : E Bje J is a solution on
{Ti-:1 11 .t
1:. (t) >t of (B. 2.1) where F(t) : E q. e J, then
0 j=1 J
‘ n n n-i
. = B. . - E a. . B.2.3
qJ J61, 1:1 1 1iJ ) ( )

 

96

I

 

 

 

m 11.t
Proof: By hypothesis y(t) : L Bje J is a solution
jifl
of (B. 2. 1) on 11 (t) >tO , therefore,
n m 11.t 11 11-1 m 11.t m (1 t
dn(EB.eJ):z,aldn_i(>_3B.eJ)+Eqe
dt j:l J 1:1 dt j:l 1:1 J
r m 11.t m r at
Since ( Z} B.e J ) z E (1 . B. e J , grouping the coefficients of
dt (.1 J j:1 J J
(.1.t

C J of the above equation results in

nu n n n-) 1‘1 't
2 (B11. - B. B aiu. - q.) 9 J ._. 0.

Since this is true for all (t) >to, the theorem follows.

Corollary B.2.3Z If Bj i 0, “j i 0 and pi i (ij for i, j : 1,2,...,m,

 

then m coefficients of (B. 2. 3) aj, are explicitly

aj+1’ ' ° ’ ’aj+m-1

related (B. 2. 5) to the remaining n—m ai's.

Proof: Consider the system of m,m < n, equations

(B. 2. 3) written in matrix form

 

Va : L
-q .
1 n n-i
l_ l_ \j‘
Wherea ”[aj’aj+1""'aj+m-1J’ “[BIJJ‘11 ; ait‘l
_qn1+ n +V‘a n—i] V‘ i‘the um over alli if’ '+l
Bn J‘Lm "S’ ipm ’: b S : .11.] w”.
1

j + 111-1 and

97

 

 

n j n-j-l n—'-1n+1
“1 “1 "‘ “1 J
V: (3,2,4)
n—j n-j-l n-j-m+1
Hm 1n m
Since det (V) : pill-J-nhkl u3_J-Jn+l . . . u:;J_m+l l I (111—11.),
m: i >j J

and by hypothesis (ii i 0 , Hi I. “j therefore det (V) ;f 0. This

implies the conclusion since,

 

 

a = v‘1 1. 03.2.5)
m
Theorem B. 2. 3: If y(t) : BO + E B. sin(w.t+¢.) is a
j--- 1 Jm J J
solution on 12 )t] >tO of (B. 2.1) and F(t)- _ q0 + 23 q. sin(w. t+9 .) then
i=1 J
(l) qo : _an Bo
n n'i (fl-1)“ ¢
nrr E are. cos( . + .)
2 .cosB. — B cos —-—+ ¢ B. . 1 2
(>qJ J JsJ. < ) 31:1 1 J

(3) q.sin9. — _.B can sin(—-—+ fl.) -B. Ea ...i.-i sin 1((1—--———J2 1)" + (5.)
J J J J jiT‘l 1J J

where j : 1,2,... ,m.

 

 

IT]
Proof: By hypothesis y(t): BO + E Bj sin(o.) Bjt+¢j ) is
i=1
a solution of (B. 2.1) on II It] >t0, therefore
dn J m n dn—i m
(B + E B. sin(w.t+¢)) : :3 a. (13+ 2: B.sin(o1t+¢j ) ) +
or“ O j=l J J 1:1 1 d1“‘1 0 jzl J J
m
+2 .sinr.t+8..
qo J21 qJ (»J J)

98

Calculating the indicated derivatives, this equation is,

m

 

 

n . nTr 1hr .
E B.'-.8111——+¢. cos .t+cos —— +¢. smut
n-l m n-i (n—i)1r (n—i)1T
: E a.B + :3 13.9.). [Sid + ¢.)coso.1t+cos( + ¢.)sinto.t]
1:1 1 0 j:1 J J 2 J J 2 J J
m
+a B +a 2 B. [sin¢. cos w.t+cos ¢.sin oo.t] +
n O n jil J J .J J
n
+ E . sin 9. cos .t + . cos 0. sin ' .t .
qo J-___1(q1 J ”J qJ J ”1)

Grouping the coefficients of sin Lujt and the coefficients of cos (11.1.

and equating each to zero results in the theorem.

m m
IfrtzB +EB.co'¢.sin'.t+23B.sin¢.cos'.t
JJ) 0 2-1 1 D J ”1 -..l J 1 "”1
J- J—
m
is the solution on I: It] >tO of (B.2. l) and F(t) : q0 + :3 qj cos OJ sinogjt +

131 J l

E q. sin Oj cos will then in a manner similar to that of the proof of Thm.

 

 

 

B.2.3
(I) qo : "311130
11 n17 n n i (n 1)1T
2 .cosB. :B.cosf).'.cos—,—- Sanson- cos —
J J qJ J J J[”J 2- ,21 1 J 2 J
- B. sine. [..i1 sinm- E aoJJ'J sin (11-1)“ ]
j j j 2 :1 1 j 1 2
n nTr n n i (n i)
3 .sinG. : B. cosO. (.sin—r- E a.w.- sin - 1T
( ) qJ J J J [...J 2 1:1 2 l
n 1 . ( .)
+ B. sin 9. [an COS 21- S 21.;31-1 5 11-1 W ]
j j j 2 :1 1 2

i .

:3? E y
' V
‘A :

Li”

arr-”rm- ~‘- .- ‘.—‘T *7.

vi trad“.- .

99

Corollary B.2.5: Ifm:1,p—q¢ 0, 2,4,...,ool£O,

 

 

 

 

 

 

 

Bl f 0 then
(I) Boan : -qo
‘lF sin((—n—_—CJJ-TL + ¢ ) - F cos ((n-q)1r + ¢ )]
(2) a I 1 2 1 2 2 1
p con-q sin (*1)-wa
l) 2
[F sin((n——-—E—1 + (J ) — F cos ((n-phr + (11 )]
q
n-q . (1)-th
o 1 Sin 2
“Jth F _ qlcosel In n17 + (J + V , Jn-i (n-i)1r ¢
ere, 1— *E—T— " “’1 C05 (’2— 1) Z dial cos(—T— + l)
qlSillel n o n" n_i . (n_i)Tr
F2 *T 0J1 SUN-‘2' +¢lJ + Egaitol sm( 2 +¢l)

:1“, IS the sum over all i. iJJP: (1°
3

Proof: This is a direct consequence of Thm. B.2. 3

 

for m :1.

coat .- tr‘ '

APPENDIX C

 

ON A SPECIAL INTER-RELATIONSHIP OF AN M-ORDER

 

POLYNOMIAL TO A SINE FUNCTION

 

The derivation of P1 1

as defined in the proof of Thm. 4. 3.1 is developed in this

,n-lJDJf1Bo + B si11(wt + (21))

 

 

 

Appendix.
m .
Lemma C.1: IfT(y):.‘_Ia.yJ,c1 ¢0andyzy +y
.130 j 111 1 2
then
where L : [(1 a . . u ] Y' : [l y y‘3 ym ] and
' o' 1’ °’ m ’ 1 ’ 1 ’ 1 ’ ’ 1
O
f. (O) o 0 o
1) 1
Jaio 111 O 0
Y2: 2 2 2 2 (C. 2)
,. l ) yl ) l l 0
2 O 2 1 2
m.m
. J '111-1Jm' Jm-2 (m) (In)
y210 V2 1; ’2 2 H14
Proof! By hypothesis
m . m . 31 .
T(y) :: aJ.<y,+y,)J : 2; aJyJZ<1+—.—- )J
Using the binomial theorem
n1 . j j! yl r

T(y): E €1.sz . , (—f—) Jr<j<m
1:0 J ZrzoJJ’r“ r' 12 " ‘

101

j 111,.
= Z aYJ '23 (J1) (--)
J—o J r:0 Y2
m .
J 1+ 1+
: Z + . . ( . +
NH J
umyz (JJJJYI

The lemma conclusion follows by writing this equation in matrix

 

form
T(y) : L Y2 Y1
where Ll, Y2 and Y1 are defined in the lemma conclusion.
m .
Lemma C.2: If T(y) : Z aij, aj £0, and y(t) :
1:0
Blsm(:;t-t+0)+BO, BO 7‘. 0, then for L : [00,a 1’ . . . ,am], 61> : out + (J,
n1/2 n1/2
T(y, t) : LRlReYe 2.1.2) szcos 2_]<I> + jTl b2j_ls‘1n(2,j-l)<1>

where m is even, Ye' : [1, sin <I>, cos 2CD, . . . ,Cos m<I>]

 

 

5 (3) o o o o “
B13.) 11) o o o
Rf 33131 B01?) o o
B%"ll%“1 BS“ (‘1‘) 5:14:13) 12:1 0 E
BM“) BS“ 1?) BS“ 53‘) 3.1.31) (:11) J

102

 

 

 

 

 

 

 

 

f— _
1 o o o 0 ‘
0 B1 0 0 0
Bit) 43513
0 0 0
2 .2 2
Bil i)
l 0 _ 0 0 0
i .a2
J C O
l
J Brln—l'iii-l m—l 11—
: "T'J 1B1 101
' 0 2m"; 0 . . . _1m—I 0
11 n1 1
Rm, m J -lelILJ) +Bm m) 1
1 m/zl 0 2 o — 1 0 J
L ”“4: 2111-1 2111—1 J
2 o
_1
111-1 1114-1
and 2 _2_.._
T(y,t) : LRlRoYo : L hzjc052j<1> + L sz_18111(2j-1)<I>
J:.0 J 1
where n1 is odd, Y5 : [1. sin<I>. cos 2<I>, . . .. sin m<I>] and R0 2 Re

with 111 '— rn -1..
0

Proof: Set yZ : BO a11dy1:sin (I) in (C ..1). It is well-

known [17, p. 82] that for k even

 

, B1112) Bk k/Z , 1‘
yk - l k/ + T1 V (—1)\ k cos 2v<I>
1 .2k 2‘1 v.1 '2" "
and for k odd
Bk ...—k“; k
1 J \ k-l

)1 1 EFT-T Z: (-1)' T _\, sin (2\'+1)<I>.

10.3

The len1ma follows by writing (C. 1) in ter111s of the
auox'o deflned quanutles (Y2: R1 and errRe Ye or ROYO).
The following, relation, is an example of Lemma C. 2.

for rn r1 5,

 

 

 

 

2 B1 3 3 2
T(Y't):[°o+a1Bo+°2(Bo+T)+a3(Bo+§B0Bl)]+ E
B [ +211 B +0. B2 3B2 ~' 05 ”$7
101 20 3(3 0+3 1)]s1n('.,1t+ ) 1
3
Bi .
-_Z_[az+3a3 BO]COS 2(wt+¢) is“;
B: 1
—a3 71— sin 3(wt + (D) .
1T1 .
Theorem C. 12 If f(y) : E Q.yJ,Q i O, and y(t) 2 B + B sin(wt+¢).
ij J m o 1
then R
an-1(D) f(BO+B151n(,Jt+¢)) 'dn-lbo + jfleijjcos ZJwt +
k r r
v T . .1 S, «.1 ._ ,1 ._ ‘
:, 1\jb2j51n 2J.,1t + .1, ij2j_ls1n(2_] l)..,t + .1, Sjb2j—1COS(2J 1)Lut
J 1 _]--l J--1
1» p (D) — Dn’l d Dn-Z d Dn‘3 d Dj~ dj p (D) -« 1
“”9 1.11-1 " ‘ 1 ’ a ‘°°°' n-l' 71:3“ ' 1’0 “ ’
r11 111-1 r11+1
k : r z— for m even, k : r : for 111 odd and.
2 2 2
. n-l .. (n—1)Tr n—l . n—i-l . (11—i—l)1T
Mj _- (23:11) cos(2J¢ +——Z———-) - 13 di(2_]w) cos(23¢+ ————5--——- )
1:1 “
11-1 (11-1) 11 l Il-i-l (ll-1-1)
N. : 4,232.0) sin(2j¢+ 1T) + :; d.(2jw) sin(2j¢ +—————1 )
J 2 1:1 1 2
. n—l .. (n-l)1r n-l . n—i-l .
LJ- = [(ZJ-l)w] COS((8J-l)¢ + -—§-—) - 3 d1 [(2)-1M] cos((ZJ-lW +
1.21

1-'—1
(121 hr)

104

 

SJ. _—_ [(2j—1)w]n-lsin((2j-l)¢ + “z ) - 2: di[(2j-1)w]“’i‘bum—1W +

(n-irl)fl

a )

Proof: The theorem follows by determining

 

P n-1(D) f(BO+Blsin(wt+¢)) which is equal to

1
u u
le_l(D) (jEOszcos 23¢ +j§1sz_l 5111(23-1) CD)

by Ler11111a C. 2.

W” ‘“"“" ’
P.
I

 

',

:11 1: 1,11

,...-"1

 

   

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