Topological analysis and synthesis of linear systems

Journal The Frankl& Institute Devoted to Science and the Mechanic Arts Vol. 274

DECEMBER 1962

No.

6

TOPOLOGICAL ANALYSIS AND SYNTHESIS OF LINEAR SYSTEMS S. BELLERT’ ABSTRACT A new method facilitates

is presented,

the analysis

namely,

of linear

posed by Wang and Woiniacki. in a general

manner

By making

The algebra linear

determination

numbers

algebraic

equations.

of all possible

of structural

of structural

is a continuation numbers

of a two-terminal

or transfer

which

makes it possible with different

to solve network.

structures,

function.

can also be applied, Moreover,

numbers

of the idea pro-

or a four-terminal

we can find a set of systems

function

of structural

algebra

This

systems.

of synthesis

computer

which realize a given impedance to solving

the so-called

The algebra

the problem

use of a digital

electrical

aside from electrical

it gives us a very simple

engineering, algorithm

for

trees of a graph or multigraph. 1. INTRODUCTION

The theory of electrical networks as an independent domain dates from the works by Kirchhoff who established the fundamental laws in 1845. Since that time several different developments of the method initiated by Kirchhoff have been set forth. These methods for the most part make use of the theory of determinants or the matrix theory of Cayley. It is worth noting that these methods were quite sufficient till the time At when systems of more complicated structure began to be considered. present, in realizing systems such as in automatic control, radio engineering and wire communication, the degree of complication is often very high. Analysis of such systems by classical methods led to extremely tedious calculations. In an attempt to simplify these methods, there appeared a whole class of new methods called topological methods. No really basic concepts of the algebraic nor combinatorial topology are involved in these methods. They are spoken of as topological, mainly because the configuration of t Communication

Department,

Warsaw

Technical

University,

Warsaw,

Poland.

(Note-The Franklin Institute is cat rcsponsiblc for the ~tatemcnts and optnions advanced by contributon

in the Jouarr~~) 4%5

S. BELLERT

426

[J.F.I.

the connections between the elements of the system makes possible the Most notable in this domain are reading of their dynamical properties. Parseval, Galiamiczew, Bayard, Foster, Okada and Mason, the initiator of signal-flow graphs. In addition to the research done by the above-mentioned authors, special credit should be given to the research of Chinese scientists initiated by Wang in 1934 (z).~ It should be noted that H. Woiniacki, a Pole, contributed independently many new ideas to the concept (6). The method of topological analysis and synthesis of electrical networks presented in this paper is based on an algebra which the author proposes to call the algebra of structural numbers, and which is a continuation of the method of Wang.

75) 4

al

3 s

2

I

I

4

4

3

5

\

5

5

!

FIG. 1.

We shall now define certain terms which will be used in the paper. The first is the determined graph. A graph will be called determined if its branches are set into one-to-one relationship with natural numbers so that each branch corresponds to another number. The function setting the branches into one-to-one relation with the above natural numbers will be called the From a mathematical standpoint, a determined graph describing junction. can be considered to be an ordered pair, A = < G, j>, where G is a graph and j is a describing function. A connected (nondetermined) graph which contains no meshes will be called a dendrite. A determined dendrite will be spoken of as a tree. ’ The boldface

numbers

in parentheses

refer to the references

appended

to this paper

TOPOLOGICAL ANALYSIS

Dec., 1g6z.l

Two trees, T, = < G,, f, > and and will be denoted by 7, - T2, ;f_fi(G,)

427

T2 = < G,, f2 >, will be called similar = f,(C,), that is,

Two determined graphs, A = < G,, f, > and B = < G2, .f, > , will be called similar and denoted by A - B, ij their expansions into trees contain exclusiuely similar trees. Figure 1 presents two graphs with similar structures and their respective expansions into trees. 2.

METHOD

OF

STRUCTURAL

Foundations of the Algebra of Structural

.kumbers

NUMBERS

A structural number is somewhat similar to a matrix, which is widely applied in electrical engineering. This similarity is, however, merely apparent, since the calculus of structural numbers is based on quite different definitions of operations from those in the matrix calculus.3 We shall call the structural number the system, A, of natural numbers arranged as:

(2)

This

system

will be considered

as a set of columns,

which in turn are unordered sets of natural numbers. The columns--as sets-will be considered as equal if they contain the same elements indeIt is assumed that in a structural pendently of the succession of elements. number no identical columns can appear. The operations on the structural numbers are shown by means of the following examples. 7. The equality of the structural numbers

Generally:

ro1umn.r.

7wn structural numbers- are .sazd tl~ he equal if they rontazn the came

428

S. BELLERT

2. The sum

[:

:

::

of the

+

[J.F.I.

structural numbers

I:

:

:I

=

[J

Generally: The sum of two structural numbers A and B is the structural number containing the columns of the number A and B, except for the identical columns (and also containing no other columns). 3. The product of the structural numbers

A

Generally: The product of two structural numbers A and B is the structural number, the columns of which are the sums (in the sense of the algebra of sets) of all the possible combinations of the columns of the numbers A and B, except for identical columns and for such columns where any element is repeated (moreover the product contains no other columns).

It call easily be shown that the operations on the structural satisfy all the principles of operations known from elementary (commutative, associative, distributive law, etc.). A structural containing no element is called the Zero number, and is denoted as

numbers algebra number

[] = 0.

(4)

From the standpoint of mathematics, the set of structural numbers which have the same number of rows forms the so-called commutative ring. This ring contains the divisors ofzero, that is, from the equality AB

= 0

(5)

it need not necessarily follow that A = 0 or B = 0. It can easily be proved that the subtraction of structural equivalent to addition, since we have

numbers

A-B=A+B.

is (6)

We also have” A. A = 0 and _4 + A = 0. We are going to discuss now the following concepts connected with structural numbers: (a) a geometrical image; (b) a complementary number and inverse image; (c) a determinant function; junction of simultaneity. 4 It is worth possessing thr othrr sumptlons

noting

that

the

one and only one empty hand,

for .i = 1, MC have

of M’ang‘s

method.

proposed

algebra

column

[O] = 1.

1.1 _ 1.

The

(d) an algebraic

contains The

equations

unity

equation

derivative;

which

ic the

structural

A.4 = 0 is satisfied

A.4 = 0 and

and (e) a

.4 + .I = 0 ai?

number

for A f the

1

On

basic

a--

Dec.,

TOPOLOGICAL ANALYSIS

1962.1

429

a. The Geometrical Image of a Structural Number. It is known from the theory of complex numbers that the image of a complex number is a point on the Gauss plane. But the geometrical image of a structural number is the graph all trees of which are determined by the columns qf that number. A geometrical image of the structural number is then the class of similar determined graphs. For example the determined graphs presented in Fig. 1 constitute the image of the following structural number A :

2

1

1

2

3

3

3

4

4

4.

5

5

5

5

5

1 I

A=

b. The Complementary Number and the Inverse Image of a Structural flumber. If a set ofelements occurring in a given structural number A is denoted by $, then the complementary structural number will be the number Ad, the columns of which are di&ences (in the sense of the algebra of sets) 6: -

c,, 6: -

c;, . .

6: - C,>

(7)

of the set 6: and the columns Ci of the number A. Example:

A=

F 1; 231

427

(8)

ild=

5 3-

4_

The inverse geometrical image of a structural number is the graph all co-trees of which are determined by the columns of that number. It can be noted that the image of a given complementary number Ad is simultaneously the inverse geometrical image of the number A.

Table I shows the images and inverse images of certain structural numbers. It may be noted that the inverse image of a structural number is a graph with a dual structure with respect to the graph representing its image (for planar graphs). c. The Determinant Function of a Structural Number. tion is determined on the set of structural numbers:

r.;1 CY,,

det A = det
The following

.

Lyz,. .

a!,”

a$”

b=l I=,

:

where z is the subset of the given complex shall write det A instead of det ‘4. %a& t ;

func-

numbers

c,,,,. For simplicity

(9)

we

430

S. BELLERT

[J.F.I.

Example:

TABLE

No.

1

I.-Geometni

Structural

Images and Irwerse Images of Some Structural Numbers. Geometric images

numbers

c

d

d I

Geometric inverse images

CL

TOPOLOGICAL ANALYSIS

Dec., 1g6z.]

43’

The algebraic d. The Algebraic Derivative of a Structural Number. of a structural number A is determined by the following formula aA -=

aa

A

derivative

columns not containing CYbeing omitted, and the element cx being omitted.

It can be noted that the following

d;t(@)

relation

= $

(10)

holds

kA].

Example:

I 23

z-= aA

jl 31

a2 aA =

41

47

42

43

1 ;

1 .

Property 1. The inverse image of a structural number of a number A in which the branch a! is erased (Fig. 2).

aA/aa

is the inverse image

FIG 2. For any two structural

numbers

A, and A, the following

;

(A, + A,)

;

(x4, . A,)

= 2

relations

hold

+ 2, (11)

= z.4,

+ 2

A,.

S. BELLERT

432

The notion of an inverse derivative, -= 6A 6a

A

6A/6a,

[J.F.I. is defined in the following

manner:

all the columns that contain the element (Y being omitted.

(12)

Example: A=

[;

:

;

The inverse derivative

t]

;

;=

[;

fl;

g=

[;

1

;]

has the following property:

2. An inverse image ofa structural number 6A/6a! is the inverse image number A in which the branch (Y is short-circuited and erased.

Property of a

It can be noted that for any two structural lowing relation holds : $

(A, + A,)

= $

numbers

A, and A, the fol-

+ 2

(13) ;

(A, . A,)

= 2

A, + 2

A, + A, . A,.

FIG.3. e. The Simultaneous Function. The term simultaneous function of a given structural number A (the inverse image of which contains oriented branches Q and 0) is given to the function denoted by the symbol Sim

‘?

and determined

_.1 (C!!! aA aff

’

ap

’

G,kf

z

as follows : 2s a linear combination

Cc+1" and (‘_I" of the terms occurring simultaneously

aA

and det i’ @

(14)

with the coeficients

in the functions

det !A; aa

Dec.,

I

TOPOLOGICALANALYSIS

g6z.l

433

2. If-after erasing from the inverse image of the number A the branches determined by the elements of a given term--we obtain a mesh with the same (d#erent) orientations of the branches (Y and /3, then that term is assigned the coejicient ‘( + 7” (” - 7 “) (Fig. 3).

Analysis of Electrical Systems by the Method of Structural

Numbers

By means of the notions of determinant function and simultaneous function, we can express any functions characterizing a system, for instance, input impedance, voltage-transfer function, current-transfer funcor composite transfer coefficient. tion, voltage-current transfer function, Let us consider a passive four-terminal network (Fig. 4).

FIG. 4

For such a four-terminal can be determined :

network

the following

characteristic

functions

(15)

In terms formulae:

of the theory

of structural

numbers

we obtain

the following

(16) det A 7 = ___‘z ‘il

;

Fe5 = In

det ?! ; aa In all the above formulae .4 is a structural number. the inverse image which is the given four-terminal network. ; IS the set of impedances tht: system. Calculation of the structural number, :1 ii c~arried our on the b,lsis the following:

of of 01

S. BELLERT

434

[J.F.I.

Theorem 1. The structural number A is equal to the product of P,, Pz, . . . , P,, of one-row structural numbers corresponding to all the independent meshes of its inverse image. Example

1.

The structural number A, the inverse in Fig. 5, is equal to the following product: A = P,P,P,

= [l 3 51

image

of which

[2 4 51

is represented

[3 4 61.

FIG. 5.

FIG. 6

Hence, [1 3 51 x [2 4 51

13 4 61 11111111333

22244555224

A=

i

34636346466

Also A = [l 3 51

[l 2 61

[l 2 3 41 ; etc.

A structural number can also be determined simply from knowledge of the graph which is its image. We make use, then, of the following: Theorem 2. A structural number A with a given geometrical image, having n vertices is equal to the product P,, P,, . , . , P,_, of one-row structural numbers corresponding to cut sets of the (n - 1) arbitrary vertices (nodes) of its ima,ge. Example

2:

The structural

number,

the image

of which

A = [l 2 3] [2 4 51 [5

is presented

6 71 [l 71.

Also, A = [3 4 61 [2 4.51

[5 6 71 [l 71;

etc.

in Fig. 6, is

TOPOLOGICAL ANALYSIS

Dec., rg6z.]

Let us now take an example of analysis of an electrical network in which we use the algebra of structural numbers.

43s four-terminal

Examfile 3: Let us calculate the voltage and current transfer functions terminal network shown in Fig. 7. We have A = [2 3 61 [2 4 51 [l 4 61.

of the four-

12

FIG. 7. We perform the multiplication

in terms of the following scheme:

12 3 61 x [2 4 51 -11 4 61 2$222333333336$6661

A= We have

1

5552224

45552

1461461

61461

dA -=

al

The columns occurring simultaneously cated by rectangles. The simultaneous

in the above derivatives function is then

are indi-

S. BELLER~

436

[,j.F.I.

Hence,

It can be seen from the above example that great advantages of calculation are gained by the use of the algebra of structural numbers in problems concerning analysis of systems. Synthesis of Electric Systems by the Method of Structural Numbers The classical methods of synthesis of linear systems are prescriptive methods. They require an individual approach for every concrete problem of synthesis. The method of structural numbers makes it possible to solve the problem of synthesis in a very general manner without imposing any Such a problem-due restrictions on the structure of the system designed. to a high degree of complication in calculations-should be solved by a digital computer. The problem of synthesis of an electric network can be split into two stages, namely: (1) topological synthesis of a graph; and (2) calculation of the values of the individual elements in the system. The term topological synthesis of a graph is understood as a procedure aiming at determining the class of structures of similar graphs which make up the geometric image (or the inverse image) of a given structural number A. From Theorems 1 and 2 cited above it follows immediately that such a procedure would lead to determining all the possible expansions of a structural number A into one-row prime factors. A structural number can, however, have no geometric representation of itself. In this connection, the following conditions are specified as necessary and sufficient for a structural number to have a geometric image: 1. The number A must have the expansion into one-row prime factors A = PIP,...

P,.

(17)

2. An arbitrary element CQ of the numbers P,, . . . P,,, may occur at most in two Moreover, the following additional connumbers Pi, P, of the product (17). ditions must be satisfied : = 1,2,.

3. P, # 5.;

i,j

. . , m;

(i # j)

4. Pi # C

Pk; k = 1, 2, . . . , m;

(k # i).

(18)

k

Otherwise we would have obtained A = 0. It can be noted that the above conditions for the existence of the image of a structural number determine at the same time the conditions for the

Dec., xg6z.]

TOPOLOGICAL

ANAL.YSIS

437

realization of a given admittance matrix of the system; such conditions have not been specified to date (Foster (4), p. 15).5 The activities connected with the synthesis of the two-terminal RLC network which should be programmed for a computer, are set up below as an example. Example:

Let us determine a family of structures of systems realizing the impedance of the RLC two-terminal network. In terms of the method of structural numbers, the impedance of the two-terminal network can be determined by means of the following formulae :

<=_.

det ?!Y aa

det E (19)

det A ’

z

I-

where A is a structural number, the image of which (in the first formula) and the inverse image (in the second formula) is the graph of the system, and Z, is the impedance of the branch which is between the terminals of our network (Fig. 8).

In our further considerations we are going to make use of the first formula which expresses the impedance < in terms of admittances y, of the branches of our network. Let us assume that in each branch of the network there will occur a series connection of the resistor, of the induction coil, and of the condenser. Thus Y* =

1

(20)

sL, + R, + s-‘C-’

Assume that the impedance, the synthesis of which is being carried is determined by means of the following real, positive function 5 After submitting

this paper for publication,

alization of the admittance

the author

matrix had been presented

was informed

in the doctoral

that the conditions

dissertation

out,

of re-

by Lee, in 1960 (9).

S. BELLERT

438 z(s)

=

[J.F.I.

U,+,J”f’+ . . . + a, + . . , + 6,s”+

a_(,+,,

. . + 6, + . . . +

b+l)

hJ:-

=-.P(s)

(21)

Q(s)

We introduce

the following notations : 6 = number of branches in the network, w = number of nodes in the network.

It can be seen that in the case of the realization of the system on the basis of the first formula (19), the degree of the denominator Q(J)is equal to (6 - w + 1); hence 6-w+l

=n.

Since, in the process of determining RLC elements of the system we obtain following inequality must be satisfied :

(22)

36 unknown values for the series (4n + 4) algebraic equations, the

36 > 4n + 4, which together with formula (22) number of nodes in the network

(23)

yields the following

evaluation

for the

(24)

(b)

(a) FIG. 9. Since the number m of the factors in the product we can generally assume that

m=E

+ k;

where E(x) denotes the entire x. In connection with the above, nodes as follows

connected

k = 1, 2,...,

with carrying

k = 1, 2, . . . . out the synthesis

1,

(25)

the network, will have the number

+k+n; The activities as follows :

(27) is equal to w -

of

(26) are specified

Dec., I cjh]

TOPOLOGICAL

ANALYSIS

439

+ 1 and consider

1. We assume that m = E

the following

set

of branches @ = (1,2,

. . . bJ;

+n+l

in terms of which we set up the products

of one-row

A = P,Pz...

structural

numbers

P,,,,

according to the principles of realizability given formerly (17) and (18), with the assumption that the numbers P, consist of at least two elements. For example : A = [l 21 [2 31 [3 4 51 . . . etc. By carrying out the multiplication we obtain all the possible structural numbers A,, A,, A, . . . Al, . . and thus all the possible graphs corresponding to k = 1. 2. We calculate

the algebraic

derivatives ail, -$

dA, -$

3. We calculate

the determinant

..’

aA, <‘

‘.’

functions Ly = 1, 2, . . . 6.

det1 A” det fiZ. r’ au 4. By comparing

aA, a(y’

the coefficients

of the rational

functions

det 5 p(s)

r aa )

=

det A,

Q(J) we

1‘

obtain the following sets of nonlinear f:;,

(R,, . . , L,, .

Yyn+t)(4, . . , L,, q;(R,,

(27)

equations

:6

) c,, . . ) = a,,,,

“.

>

c’,, . . . ) =

~_(,,,),

. . . . L,, . . . . C,, . . . ) = bn,

----------_-

cp9-L . ..> L,,

(28)

_----_-_----_

. . , C,,

. . ) = b_,.

6 We disregard such structural numbers as .41,for which the numerator and denominator 01 the expression (27) are divisible.

S. BELLERT

440 TABLE

II.-Flow

Chart for a Digital

Computer for Synthesis of an RLC

[ J.F.I. Two-Terminal

Network with a Minimal Number of Branches.

These sets of equations can be solved by a digital computer, for example, by means of the relaxation method; we obtain then the values of the

Dec.,~gfk.]

TOPOLOGICAL ANALYSIS

44’

elements in the particular branches of the network, the structures of which have been determined by the structural numbers A,, A,, . . . , Ai, . . . It can be seen that the number of unknowns is greater than the number of equations by the quantity 6 :

In this connection the values of 6 elements in the system (for example, inductances) are assumed to be zero. 5. If necessary, we repeat the entire cycle of calculations for + 3,

etc.

It is clear that the systems now obtained will contain a greater number of branches than did the systems calculated formerly for k = 1. Table II shows a computer flow chart which includes all the operations connected with carrying out the synthesis of an RLC two-terminal network with a minimum number of branches. In the case of synthesis of a four-terminal network we make use of one of the formulae (16) for example,

The procedure

in this case will be very similar

to that

presented

above.

APPENDIX 1. Proofforformulae

( 16)

Let us prove the correctness of the formula. determining This transfer function-for the four-terminal network presented the following formula Z2 K,=_=

c

12” -

transfer function li;. be found in terms of

‘21r P

y 4

c

the current in Fig. 4--tan

4

where I,, are currents Rowing in the four-terminal network through the individual taining the branches LY= 1 and B = 2 which have the same orientation. Zzr are currents flowing in the four-terminal network through the individual taining the branches a and @which have not the same orientation. On the other hand, then the columns

circuits

con-

circuits

con-

since the columns of the number A determine all the co-trees in the system, dA dA of the numbers aa and determine those branches in the system, the reaa

S. BELLERT

442 moval of which converts bers g

and ‘$ determine

the system

into a network

then all those branches

with one cycle. the removal

the branches o( and @,that is, leads to the circuits of the currents Moreover, making use of Maxwell equations we get E;J

= 1

o;J

> 1’

[J.F.I. Identical

of which

columns

of the num-

leads to cycles containing

lzV and I+.

j = 1, 2,

, n.

(Al 1

We can easily find that ‘2 42 = 7 1 = A-’ 11

’

where A,t and A,2 are the corresponding subdeterminants of the system (Al). Clearly, A,, is the sum of all the values of the co-trees if the system with the branch erased. Thus A,t

1 = 01 being

aA

= det -, z acY

where A is a structural number the inverse image of which is the given system. A,, is a linear combination with the coefficients “+l” and “- 1” of such co-trees of the system with the erased branch (Yand 6, the removal of which implies K z 0. After removing any of the above co-trees, the system is reduced to a cycle constituting the circuit of the current Z2”or Z2,,. Summarizing

our reasoning

we state that

where < is the set of impedances remaining formulae (16).

2.

Proof

forTheorem

of the system.

Similarly,

we can prove the correctness

of the

1

It can immediately be seen that the theorem if satisfied for a system with the structure of a cycle. It is also satisfied in the case of a system with two cycles. In fact, such a system can always be simplified to the form shown in Fig. 90 or b where 1, 2, and 3 are the sums of the corresponding elements of the system. In the case of the system (a) we have

A= and then, in fact, A = [l 21 [l 31 Also in the case (b) A=

It can easily be proved that the above results and 3 consist of any number of elements.

=

Ill PI.

are correct

in the case where

the branches

1, 2

Let us now assume that Theorem I is correct for a system with n independent cycles. By reasoning analogously as above we can easily prove that the theorem is correct for a system with (n + 1) cycles. Thus, by the principle of induction, Theorem 1 is valid for a system with any number of cycles. Theorem 2 can be proved in a similar manner.

Dec., 1y62.]

3. I~roofforformulae

TOPOLOGICAL

ANALYSIS

443

( 19)

Let us present a proof for the second formula of (19). Therr 1s a well-known impedance of a network measured between the nodes d, 6 is equal to the ratio

Mhere A is the Ar,, LS the (1and ~foreover,

main determinant of the impedance matrix of the given network. and main determinant of the impedance matrix of the network with b. we have (for the system presented in Fig. 8) A = det A;

?

theorem

short-circuited

that

thr.

nodrs

A,, = ..&ddet - 3 ; 6u

where the formula for Aob results from the property 2. In a similar manner, we may present a proof for the lirst formula

of (19)

REFERENCES A&s des Colloques Phzloso(1) S. BELI.EKT, “La formalisation de la notion du systeme cybernetique,” phiques Intemationaux de Royaumont, Paris, 1961. (2) K. T. WAUG, “On a New Method for the Analysis of Electrical Networks,” Not. Res.Inst. Eng. Academia Sinica, Memoir No. 2, 1934, pp. l-1 1. (3) W. L. C~zow, “On Electric Networks,” J. Chimve Math. Sot., Vol. 2, pp. 321-339 (1940). (4) R. M. FOSTER, “Topologic and Algebraic Considerations in Network Synthesis,” t’rr,r. $mp. ,Vodern :Lktwork Synthm, 1952, pp. 8-18. (5) w. s. Pmct\..4t, “The Solution of Passive Electrical Networks by hleans of Mathematical Trees,” I’m. ILL’, Ser III, Vol. 100, pp. 143-l 50 (1953). (6) H. WOihlACKI, “Obliczanie i analiza sieci elrktrycznyrh metoda wielomian& charakterystycznych,” 4rchmxun Elektrotechn., Tom X, zeszyt 1. pp. 57-99 (1961). (7) Y. H. Kr.. “Resumt of Maxwell’s and Kirchhof’s Rulrs.” JOIX. FHANKI.IN Ihsl., Vol. 253, I’[‘_ 211-224 (1952). (8) R. B. Astr, “Topology and the Solution of Liwar Systems.“,Jo~~~. FU\KI IR’INS.I., Vol. XX, pp. 453-463 (1959). (9) H. ‘I‘. LFI:, “Graphs, Network Discriminants and Positive Homogeneous hlultilinear :\lqebralc Forms.” D.E.E. Dissertation, Polytechnic Illstitute of Brooklyn,,Junc, 1960.