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OF NONLINEAR RESISTIVE NETWORKS

ELECTRICAL ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirement for

the Degree of

MASTER OF SCIENCE

ACKNOWLEDGEMENTS

I am deeply indebted to Professor Kwong Shu Chao for his

direction of this thesis and to the other members of my committee.

Dr. Donald Gustafson and Dr. A. K. Mitra, for their helpful

criticism. I am also happy to thank my wife and my family for

the constant encouragement throughout this study.

UL

BY A RELATED DIFFERENTIAL EQUATION

2.1 Introduction 4

2.3 Existence and Uniqueness of Solution Curve . 7

III. MULTIPLE SOLUTIONS AND SOLUTION CURVE DEFINED

BY A CONTINUOUS PIECEWISE-LINEAR FUNCTION

3.1 Introduction 9

3.4 Starting Point 21

3.5 Numerical Algorithm 23

JACOBIAN MATRICES

4.2 Singular Jacobian Matrices Problem 32

V. APPLICATIONS

Network 36

of Multivalued Resistive Nonlinear Networks . . . 41

VI. CONCLUSION 45

3.2. The Ebers-Moll Model of a PNP transistor 11

3.3. A solution curve determined by Katznelson's algorithm 13

3.4. A solution curve crosses the boundaries 19

3.5a The circuit of example 3.1 24

3.5b The piecewise-linear approximation of function h for

the tunnel diode 25

3.9a The characteristic of a nonlinear element 28

3.9b The characteristic of a nonlinear element 28

3.10. The solution curve of example 3.2 29

4.1. The solution curve hits a corner 31

4.2. The solution curve enters a singular region 33

5.1. The circuit of example 5.1 38

5.2. Output voltages for different starting podjits 39

5.3. Output currents for different starting points 40

5.4. Multivalued input-output plot for the circuit in

example 5.2 43

example 5.2 44

INTRODUCTION

One of the most basic problems in the area of nonlinear circuit

analysis is the determination of the solutions (equilibrium points)

of nonlinear resistive networks. Research in nonlinear resistive

networks serves as a prerequisite to the understanding of general

nonlinear circuits and systems. This problem is important not only

from a circuit theoretic point of view but even more so from a com

putational point of view. Indeed, an essential part of most nonlinear

transient analysis computer programs is a DC analysis subprogram

[l-3]. In addition, similar problems are encountered in many related

fields such as mathematical economics, flow networks, power systems,

and least-square approximations.

Much work has been done in obtaining multiple solutions of

resistive nonlinear networks [4-9]. Some effort has been devoted

to using the technique of piecewise-linear approximation of a non

linear function [10-16]. For example, in the analysis of a elec

tronic circuit, diode and transistor characteristics can be repre

sented by continuous linear segments. This often gives considerable

insight to the problem and yields quick solutions. In general, by

peicewise-linear approximations, a nonlinear resistive network can

be characterized by the equation

f(x) = y (1.1)

where f(.) is a continuous piecewise-linear mapping of the real

2

n-dimentional Euclidean space R into itself, x is a point in R and

represents a set of chosen variables of a given network and y is an

arbitrary point in R and represents the inputs. Usually, f(.) is

expressed in the form

f(x) = J ' x + w^^^ = y i = 1, 2, , N (1.2)

where J is a constant nxn matrix called the Jacobian matrix and

w is a constant n-vector, both defined in region R . The whole

space R is divided into a finite number N of polyhedral regions

by a finite number of hyperplanes.

Although various techniques have been deveolped for solving

piecewise-linear equations, relatively little has been done in

locating multiple solutions. It is the purpose of this thesis to

develop a systematic search technique for obtaining multiple soluti

ons of (1.1). The continuation method devised by Chao, Liu, and

Pan [4J plays an important role in the scheme developed in this

report.

For convenience and to make the present thesis self-contained,

the continuation method [4] is outlined in Chapter II. In Chapter

III, the general theory of the new algorithm is developed with two

assumptions, namely, all Jacobian matrices may have arbitrary signs

but are not singular and the solution curves never hit a comer.

The dynamic behavior of the search curve is discussed. Approaches

showing how a solution curve can be reached are presented. Formu-

lation of equation (1.1) is also briefly introduced.

In Chapter IV, the corner problem and the problem of singular

Jacobian matrices are considered. Applications to other fields are

discussed in Chapter V. These include AC nonlinear resistive network

analysis and driving point or transfer characteristic curves of mul

tivalued nonlinear resistive networks. Finally, some concluding

remarks are given in Chapter VI.

CHAPTER II

BY A RELATED DIFFERENTIAL EQUATION

2.1 Introduction

g(x) = y (2.1)

or f(x) = g(x) - y = 0 (2.2)

where f or g is a continuous differentia le function from R onto

itself, X and y are both n-dimensional vectors to represent a set

of chosen variables and the inputs respectively.

A systematic search method has been developed by Chao, Liu

and Pan for locating multiple solutions of (2.2). For convenience,

this method is outlined here.

2.2 Iterative Solution Method

Consider a system of differential equations of the form

^ f.(x(t)) = -f,(x(t)) , f,(x(0)) = 0 i = 1, 2, , n-1 dt 1 1 1

where the initial conditions in f. are such that the starting point

X must lie on a space curve L of intersection f.(x) = 0 , i = 1, 2,

- - -, n-1. Since f is not a function of t explicitly, -r— can be dt

written as

dt ax dt "

and i n the x-space (2.3) reduces to a d i f f e r e n t i a l equat ion

X = J""^( -f , , - f , . , - f , , ±f ) ^ , iP ^ x(Q)e L (2.5) i ^ n—i n

where T denotes matrix transpose. Solved by the Zuler intsgraticn

method, equation (2.5) reduces to an iterative scheme

\ ^ l ' \ ^ hJ- x )( -f (x ), - - -, -f .,(V> ^n^V )'

x^ L, k - 1, 2, (2.6)

where h is the step size. The transistion in sign should occur at

the points where the Jacobian J changes sign and the solution points.

The solutions of (2.3) in the f-space are given as

f,(x(t)) = f^(x(0))e'^ = 0 i - 1, 2, , n-1

0 A. These solutions show that for any x * x(0) L, the corresponding

trajectory x(t) from (2,5) remains on L. Theoretically, the signs

of f, for i » l , 2,---,n-l are not important, because the

corresponding trajectory will remain on L so long as f.(x(0)) = 0,

for i = l , 2 , - - - , n-1. But in computational practice, t"r.e f. 1

at each successive iteration may not 'oe precisely zero; the signs

of f. for i = 1, 2-, - - -, n-1 are therefore kept negative to ensure

that the computed trajectory x(t) does not stray too far from the

solution curve L.

Depending on whether the minus or plus sign is being used,

the function f is forced to be attracted to or diverge away n

from zero along L. To ensure the continuation of tracing along

the solution curve L, the sign of f has to be changed at the

solution points of (2.2) and the points on L where the Jacobian

changes sign. The key point lies in a relation between the

determinant of Jacobian J and the directional derivative of f n

in the tangential direction of L

f = ^n = s^(Vf ) (2.8) ^ 3L~

where s is the unit vector in the tangential direction of L

and Vf is the gradient of f (x). The following theorems [4]

determine the conditions of the sign change of f .

Theorem 2.1; If f'(x) is the directional derivative of f (x) n n

in the tangential direction of L at x defined by

f,(x) = 0 , i = 1, 2, n-1 (2.9)

and A., is the (ij)th cofactor of the Jacobian matrix J of

f = (f^, f^, , f^)^, then

f' = det J / I I V I I n

where ||.|| denotes the Euclidean norm and

T ' = ( nl- ^ 2 - 'nr?

Theorem 2.2: The directional derivative of f^(x) in the tangential

direction of L changes sign if and only if the corresponding

Jacobian of f on L changes sign.

Other aspects concerning the convergence of the method and

starting point problem are also detailed in [4j.

2.3 Existence and Uniqueness of Solution Curve

Success in finding all multiple solutions depends hea-vily on

whether (2.9) defines a unique simple curve L. If L is a simple

curve, i.e., a continuously differentiable curve which does not

intersect itself, then a complete tranversal of it enables one to

find all the multiple solutions. On the other hand, if L contains

several branches, then application of the method may lead to those

solutions that lie on the branch containing this starting point only.

The following theorems gives some sufficient conditions for the

existence and uniqueness of the solution curve [9].

Theorem 2.3: Let f be a real function of n real variables and of

class C . If

and (ii) Lim | | f (x) | | = «, as x -»- °°

then any (n-1) equations off. = 0 , i = l , 2 , - - - , n defines a

unique continuously differentiable space curve.

T Theorem 2.4: Let h(x) = ( fj^(x), f2(x), , f^-l^^^ ^ ^® °^

class C . If there exists an (n-1) vector

x_ = ( ^v ^2' - -' \-l' \+l' - -' \ ^^ ^" """ k, 1 <. k < n,

such that

-k (ii) Lim | |h(x) [ | = «, as x , ->• °°

then h(x) = 0 defines a unique simple curve L.

8

Definition: Let U C R • An n x n matrix A(x) is said to be almost

positive definite on U if A(x) is positive definite for all x

in U except at most a set of isolated points on which A(x) is

positive semidefinite.

Theorem 2.5: Let f:R ^R be a C function. If J(x) is almost

positive definite on R , then any (n-1) equations of f. = 0,

i = 1, 2, , n define a unique continuous space curve.

Theorem 2.6: Let f:R->R,n?^2, b e a C map. Let

S = {x^R^I det J(x) = 0} and T = {x € R^ j x e S}

If (i) det J(x) > 0 (or det J(x) < 0 ) for all x^T

and S is at most a set of isolated points,

(ii) limi I f (x) II = °° , as | | x | | - «

then any n-1 equations o f f . = 0 , i = l , 2, - - - , n define

a unique continuous space curve.

CHAPTER III

BY A CONTINUOUS PIECEWISE-LINEAR FUNCTION

3.1 Introduction

In chapter II a systematic search method for solving multiple

solutions of (2.2) is outlined. Considerable computation time is

needed to completely traverse the solution curve, and the success in

finding all the multiple solutions depends heavily on whether or not

the set of equations f.(x)=0, i = l , 2 , - - - , n-1 defines a simple

curve L, i.e., a continuously differentiable curve which does not

intersect itself.

Instead of sol-vrLng a system of nonlinear equations, the piecewise-

linear approximation is obtained for all the nonlinear elements in

a network. As a result, a set of piecewise-linear equations (1.1)

is to be solved. A method due to Katznelson ElO] was originally

applied only to the networks with 2-terminal elements which are

strictly monotonic. Fujjsawa and Kuh [ll] showed that if f is homeo-

morphic then the algorithm due to Katznelson always converges. Here

a continuous function f is said to be homeomorphic from R onto

itself if and only if the equation f(x) = y has a unique solution for

all y.

Kuh, Ohtsuki, Fujjsawa Il23 have shown that as long as all

the Jacobian matrix determinants det J , i = l , 2, - - - - , N

10

in (1.2) have the same sign, there exists at least one solution to

the equation f(x) = y and the Katznelson's algorithm also converges.

This property is refered as the sign condition. Kuh and Chien [l3]

removed the restriction of the sign condition and demonstrated that

if all the Jacobian determinants in the unbounded region have the

same sign, the equation (1.1) has at least one solution. Chien has

also developed an algorithm that the domain of the x-space is divided

into simplices and the beha-vior of the system is approximated by a

linearized system in each simplex [l3]. Chua [l5] also introduced an

iterative scheme for finding multiple solutions. However, the method

is not suitable for networks with a large number of nonlinear elements.

In the following sections, a formulation of (1.2) and a systematic

search method for obtaining its multiple solutions are presented,

where the Jacobian matrices can have abitrary sign. A theoretical

basis for the development of the method is also given.

3.2 Piecewise-linear Approximation

obtaining the piecewise-linear approximation. If the characteristic

of a nonlinear multiport is given by

z = h(x) (3.1)

then h(x) = h(xQ) -H | | | ^ ^ ^ ( X - X Q ) = J ( X Q ) X + w^ 3 ^

where J(x^) = T~| ^ cl w is a constant vector. (J dX I XFX_ U

II

In the case of 2-terminal elements without coupling, this

reduces to the familiar tangent approximation. The Jacobian at

X- denoted by JCX-.) becomes the slope of the nonlinear function

at X-. and w. becomes the vertical axis intercept of the approximate

line segment. If the network contains n nonlinear resistive

elements, each characterized by an n -segment v.

there are N distinct segment combinations where n

i. curve, then 3

Consider a typical resitive nonlinear network as shown symbloicailv

in Fig 3.1 where the n nonlinear elements have been extracted and

are connected across the ports of a linear n-port network. As

indicated by So [17] and Chua [l5]. Fig 3.1 can be completely

characterized by an n x n hybrid matrix A and an n x 1 source vector

u, namely

Ax + u C3.4}

where the components of the n x 1 port vectors 2 and x can be any

combination of port voltage and port current pro-vided that the

corresponding components of 2 and x have the same port number.

N

Linear

Resistive

Network

rig. 3.2 The Toers—Moll model of a ?">!? transistor

12

If each nonlinear resitive element is either voltage controlled of

current controlled, then (3.4) can be written in the form

z(x) = Ax + u (3.5)

T n

z(x) = (z.(x), z„(x), , z (x)) is a nonlinear mapping from R

into R and z.(x) represents the nonlinear characteristics of the

ith nonlinear resistor. In either case, the terminal constraint

imposed by n nonlinear resistors can be expressed by its piecewise-

linear approximation, which is substituted for z in (3.4) or (3.5).

But the piecewise-linear approximation of multiterminal

elements with coupling is much more complicated. Fortunately, most

of coupling elements are of the type that can be expressed as sums

of nonlinear functions V7ith simple variables such as n

z.= E h..(x.) j = 1,2, , n -J 1 = 1 -•

= J^h'..(x.„) + «0 (3.6)

wh ere h'. .(x.^) is the slope of the function h.,(.) at x. ji lO Ji lU.

by ^e = ES< ^ T - » -"r^Cs( ^ - '^

For example, the Ebers-Moll model of a PNP transistor is represented

eb/v^ , r r ^ cb/v^ _ ^

The equivalent circuit model is shown in Fig (3.2) where

^22 = ^CS( ^^"^'"^ - ' ^ ' ^22(\b> ^'-'^

13

Therefore, by specifying the piecewise-linear approximation

to all the nonlinear elements in the circuit, it is clear that

within each region the equation is linear and is of the form

f(x) = J^^^x + w^^^ = y i = 1, 2, ,N (3.10)

where J is a constant nxn matrix, y is the input vector.

The domain of f(.) in R is divided into N polyhedral regions

by a finite number of hyperplanes (or boundaries). A typical

boundary in the x-space can be characterized by the equation

n^x = 0 (3.11)

where n is the normal vector of the hyperplane (or boundary)

and c is a real constant. In (3.10), for example, if x is the

node-to-datum voltage vector and y is the input current source

vector, then J is the node admittance matrix.

3,3 Dynamic Behavior of Solution Curves

Let R , R be continuous regions with a common boundary charac-

14

T

terized by the hyperplane n x = c. Since f is continuous piecewise-

linear, there exists an n-vector r such that

j(l) . j(j) = ,J (3 3)

where J and J ^ are Jacobian matrices of adjacent regions R

and R respectively.

f(x) = J ^ x + w^ = y, i = 1,2, , N

where y is a given input. Fig. 3.3, a continuous curve in the

x-space is determined, based on Katznelson's algorithm

x^(X) = x^ + AJ ^ (y - y= )

where

and X is a scalar parameter, If X = 1, then

i/iN 1 ^ r(l) f T(i) 1 1\

x ( l ) = x + J (y-J x - w )

= J ^ (y-w^).

It is clear that x (1) is a desired solution if it happens to be

in R" . Otherwise, the value of X has to be determined such that

x"''(X) lies on the boundary of R and R . Such a value of X is

denoted by X^. Define x^ = x^(X^) and y" = f(x ), the line

segment connecting x and x is then a portion of the desired

solution curve. We may extend the solution curve beyond x into

R""" in the similar manner.

15

ohtsuki, Kuh and Fujjsawa [l2] showed that if det J and

det J -" have the same sign, the solution curve in the x-space will

indeed enter R . And the solution curve will never enter a region

which has been traced if all the Jacobian matrices determinants

have the same sign. If we remove the condition that all Jacobian

determinants have the same sign, the problem of the continuous traverse

of the complete solution curve becomes much more complicated. Kuh

a d Fujisawa [ll] have derived a equation on the boundary

det j(i>n^j("-l= det A^^'K'"/'^'^ "^ (3.13)

This equation gives a theoretical basis of the development of the

new algorithm presented here.

In what follows, a new algorithm for locating multiple solu

tions of (1.1) is developed. The solution curve is defined and

the local behavior of the solution curve across the boundary is

also investigated. It is assumed that det J ^ 0 and the solution

curve does not hit a comer, but det J may have arbitrary sign.

The comer problem and singular Jacobian problem will be discussed

in next chapter.

Let gCx) = fCx) - y = J ^ x + w^ - y, i = 1,2, , N (3.14)

It is of interest to locate multiple solutions of g(x) = 0. Instead

of applyign Katznelson's algorithm directly to (3.14) starting

from a point selected in one way or another, the iterative scheme

i+1 i,,i..(i) . . i. , iv / i\ . / INNT X = X +X J (-g (x ), -g2(x ), , -gj . Cx ), *g^(x ))

16 x^ € L, i - 0 , 1 , 2 , (3.15)

i s used where the s t a r t i n g po in t x must l i e on a space curve L of

i n t e r s e c t i o n g . (x) = 0 , i = 1,2, , n-1 and X^ > 0 i s the l a r g e s t

p o s i t i v e number such t h a t x and x are in the same reg ion .

0 1 2 As long as X i s chosen to l i e on L, the i t e r a t i o n po in t s x , x ,

~ — remain on L.

Lemma 2.1: If x lies on a space curve L defined by g (x) * 0

1 2 i = 1,2, , n-1, then the successive points x , x , resulted

from (3.15) and the.'line segments joining these points also lie

on L.

Proof: Without lo s ing g e n e r a l i t y , assume x € L i s in region R .

Consider the following d i f f e r e n t i a l equat ion in R

4 r g . ( x ( t ) ) = - g , ( x ( t ) ) , g , (x (0 ) ) = 0, i = 1,2, , n - l d t 1 1 1

4 - 3 ( x ( t ) ) » ± g ^ ( x ( t ) ) , g„(x(0)) » g^^ x(o) = x°c . L (3.16) dt n n n no ,

3y the chain-rule and Euler integration formula,

1..0, 0 ,0,(0)-l, , 0 , , 0 . . 0. , 0. • X (X ) - X + X J^ {~%^{-3i )> -g2(^ )» > "^n-l^*"^ - ' -^n^^ -

(3.17)

where 0 < X < X , X i s the l a r g e s t p o s i t i v e cons tant such t h a t — — m m

x^(X°) i s s t i l l in R^, i . e . x^(X^) l i e s on the boundary of R and m 01

R . The s o l u t i o n of (3.16) in g-space i s given by

g ^ ( x ( t ) ) - g^(x^) e ' ^ » 0, i « 1,2, , a-1

g _ ( x ( t ) ) » g^n^-^. (3.18) o. n 'nO"

This shows t ha t i i x £ L, the corresponding s o l u t i o n cur-ze x ( t ) ,

1 0 i . e . X (A ) in ( 3 . 1 7 ) , remains on L. Following tne same procedure ,

17

in region R , x (X ) is used as the starting point for the next

2 1 iteration. Then it is clear that the line segment joining x (X )

m

and X (X ) remains on L. m

In general, if x = x (X ) is on L, then by the iterative

scheme (3.15), x = x (X ) and the line segment joing x"""

and X also lies on L. By deduction. Lemma 1. is proved.

Let the line segment joining x and x be denoted by L..

Then in fact, L = L^ U L^ U . The minus sign of g for i =

1,2, , n-1 is used for reducing the computation error in each

successive iteration.

Let d^ 4 J^^^"^-g^(xS, , -gn_i(xS, ±g^(xS)^, i = 0,1,2,

(3.19)

Then that the solution curve L passes through each region R ,

i = 0,1,2, in the direction of d where the positive or

negative sign before g (x ) in each region can be determined by n

the following two lemmas. It is assumed that the solution curve

L passes through R in the direction of d then enters region R

at X and traverses R in the direction of d

Lemma 2.2: If there exists no solution in region R and (i) the

determinants of Jacobians J and J have different signs,

then in region R the sign of g should be changed: (ii) if on

the other hand, the determinants of J and J have the same

sign, then in region R the sign of g should not be changed.

18

* * I 1

Proof: Since x and x are on the solution curve L and there * ' * I 1

exists no solution in R , then g (x ) and g (x ) must have the

same sign. Without loss of generality, assume -g (x ) is used

for obtaining d"*" in R . As shown in Fig. 3.4, T 1 T 1+1

n d > 0 and n d > 0 (3.20)

are satisfied on the boundary between R and R where n is

the normal vector of the boundary. Substituting d into (3.20)

(3.19a)

Since x is a point on the solution curve L, (3.19a) becomes

^Tj(i)"l[ 0, 0, , -g^(xS]^ (3.19b)

From the condition of (3.13)

det J n J = det J n J

it is clear that

det J ^ n^J^^^'^O, 0, - -, -g^(x^)]^

- , , i, , , i+lxij .T(i+1) T^ (i+l)"" r n n f i+l^^T = {g (x )/g (x )}detJ' ' J ' L 0, 0, - -, -g^(x )J

(3.13a)

Now, if det J and det J have different signs, then

from (3.13a) n^J^^^ [ 0, 0, , -g^(x^)]^

, T,Ci+l)'''"rn n / i+l^lT and n J^ [0, 0, -—, "gj (.x )J

T i+1 have different signs. To assure n d > 0 on the boundary,

the sign of g should therefore be changed.

19

On the other hand, if det J and det J have the same

sign, then n^J^^^ [0,0, , -g^(x^)]^

and T_(i+1) n J

-1 [0,0, — . -g^(x^"'^]^

will have the same sign and

,i+l ,(i+l)~ r^ f. f i+lxlT d = J L0,0, , -g^ (x )J

T i+1 should be used to assure n d >0 on the boundary. Similarly,

if +g (x ) is used for obtaining d in R the same conclusion result

Lemma 2.3: If there exists a solution in region R , then the

transition in sign of g should occur if det J and det J o °n

have the same sign of g should not occur if det J and det J

have different signs.

Proof: It is clear that g will change sign when the solution

cuirve L passes through a solution point. Hence g (x ) and g (x )

should have different signs.

20

From the same argument as that given in Lemma 2., and

conditions

Tji - T^i _ n d >0, n d >0

.r.A A . ,{1) T Al)-1 . , .(i+l) T ^(i+l)-l and det J n J = det J n J ' ,

Lemma 2.3. is proved immediately.

In view of the above, the solution curve can easily be

taken care of at a local point on the boundary when Jacobian

determinants have arbitrary signs. Therefore the iterative

scheme of (3.15) allows continuation of tracing along the solution

curve L and, as a result, all the solutions on L can be obtained.

If g.(x) = 0,1,2, , n-1 defines a unique simple curve L,

that is, one with only a single branch, the L passes through

all the solutions of (3.14). Once a starting point on L is

obtained, all of the multiple solutions of (1.1) can be located

by using the technique described in this section. The following

lemma gives a sufficient condition for the existence and unique

ness of this solution curve.

Lemma 2.4: Let g(x) be a continuous piecewise-linear mapping of

R^ into itself and let J ^ denote the matrix by deleting the

kth row and nth column of Jacobian matrix J of region R , and

let J , denote the matrix consisting of the first m rows and km

first m columns of the constant Jacobian J, . If there exists

a k, l<JiC^n, such that for each m = 1,2, , n-1, J, , J

21

km • "km ' ""' (N)

J j do not vanish and have the same sign then g.(x) = 0, j = 1,

2, , n-1 defines a unique simple curve.

Proof: Let h(x) = (g^(x), g2(x), , g _;L '' ^

and x_^ = (x^, x^, — , x^.i^x^+i. — , x^)^

If the value of x, is fixed, then the mapping h of R ~ into itself

is still a continuous piecewise-linear mapping. For any x, ,

h(x_ .,u) = \ x_^ + v^(u) - y_^, i = 1,2, , N

where y_^ = (y^.yj, — , yi . , 7^+1. — . y„)^

Since for each m, m = 1,2, , n-1, J, , , J, do not Icm Km

vanish and have the same sign, the mapping h with fixed x, is

a homeomophism of R onto itself. This implies that for any

given input y and any number u of x, there exists one and only

one point T

x_j (u) = (x^(u), X2(u), , X^_;L^^'^' ^+1^^-^' n - ^

such that h(x i,u) = 0. This guarantees the existence of a unique

continuous piecewise-linear function F: R ->R for h(x) = 0 such

that X ,= F(x, ). Hence F(x,) defines a unique simple curve L(x, )

in R for all

A starting point lying anywhere on the intersection L of

g.(x) = 0, i = 1,2, , n-1 has to be found before the root-finding

technique described in the previous section is applied.

22 Let M be a positive number such that for any x » (x., x-,

T , X ) with X, >_ M, X is in an unbounded region. The union of

such unbounded regions is denoted by U. An arbitrary point x in

k k

R is chosen such that R U. With a fixed value of x, = u,

u >.M, a point x = Cx^(u), , Xj^_^(u), u, x^+^(u), , x^(u))^

which s a t i s f i e s g^(x) - 0, i » 1,2, , n-1 i s searched i n the (n-1) dimensional space R

Let g(x) be a continous p i e c e w i s e - l i n e a r mapping of R onto

i t s e l f defined i n ( 3 . 1 4 ) , and l e t J ^ ^ denote the matr ix by

d e l e t i n g the kth row and nth column of Jacobian mat r ix J of

the reg ion R .. In the following a lemma which was proved by

Kuh, Ohtsuki , and Fuijsawa Cl2] i s s t a t e d .

Lemma 2 . 5 . Let g be a continous p i e c e w i s e - l i n e a r mapping from

R i n t o i t s e l f defined by ( 3 . 1 4 ) . For an a r b i t r a r y v, i f det J

i = 1,2, ,N have same s i g n , then the re e x i s t s a t l e a s t one

s o l u t i o n X to the equat ion g(x) = 0. An immediate consequence of

Lemma 2.5 • i s as fo l lows.

Coro l l a ry : I f f o r a given k , 1 _<k<_ n , such t h a t in each region

R- , R" c ,^» ^ ]. <io ^ot vanish and have the same s ign then for

any x, , x, >^M, there e x i s t s a t l e a s t one s o l u t i o n for

g^(x) - 0 , i « 1, 2 , , n - 1 .

23

This corollary gives a sufficient condition for the existence

of starting point x . The convergence to x from an arbitrary

initial guess in R-', R-'c U is guaranteed by the Katznelson's

algorithm.

in the following steps.

Algorithm A:

Step 1. Find an initial point x in an unbounded region R according

to section 3.4. Set j = 0.

Step 2. Compute g(x ).

Step 3. If j=f 0, compute d according to Lemma 2. and Lemma 3.

^ 1- . ^0 .(O)"-'-, , 0, , 0.,T Otherwise, compute d = J ( -g,(x ) , - - - , -g (x )) .

i i i * Step 4. Compute x = x* + d . If x is in R then x = x is a

solution.

Step 5. Compute x-' = x" + X~ d , where X-" > 0 is the maximum value • • * * *

such that x(X) = x-"' + Xd" , 0 <_ X <_ X^ is in R^. If X^- » go

to step 7.

Step 6. Otherwise, identify region R-" . Set j = j +1 and go to

step 2.

Step 7. Set j = 0. Repeat step 2. to step 6. except in step 3. that

if j = 0 compute d° = J^°^ ( -g^{^^) . . +g^(x°))'^

gand in step 5. that if X-' ->•=», stop.

Two examples are given here for illustration,

24

Example 3.1. Consider the nonlinear circuit of Fig. 3.5a, consisting

of two tunnel diodes and two constant current sources and a linear

resistor. The piecewise-linear approximation of function h for the

tunnel diode is given in Fig. 3.5b with

' 2x X < 3

1.5x-6.5 x > 5

8^^\x) J ^ ^ ^ + w^ - y 0, i - 1, 2, , 9

are obtained. All regions are shown in Fig. 3.6. The equation

associated with each region is given as:

g^»Cx) 3

25

1= 2] +

-6 .5 '

-6 .5

In this example, y, i.e. the input current source is given by

y =

13.5

13.5

h(x)i

h for the tunnel diode

26

R3

R2

R1

R^

R5

R*

R '

R3

R^

X

Fig. 3.6. The regions in the x-space

If the solution curve L of g,»0 is traced, then the points

on L with g^» 0 are the desired solutions. For x.. >, 5, in each

7 8 9 unfaoxjnded region R , R , R oo"* 2.5 has the same sign and does

not vanish. Hence from Lemma 5, for a given value of x. = 5.5 > 5

0 T a corresponding solution x » Cy'5, 6.25) is found by solving

7 8 9 0

g_. Cxi = 0 in the unbounded regions R , R , R . Using x as a

starting point, the search along the curve L of g =0 leads to

five solutions C5.714, 5.714)^, (6.316, 4.211)^, (7.154, 2.115)*,

(4.211, 6.316)^, C2.115, 7..154) .

In fact, a simple solution curve does exist in this example

even though Lemma 4, is not satified. Similarly, if g O is

traced, the same five solutions are obtained. The solution curves

and all solutions are shown in Fig. 3.7.

27

IG.OO

F i g . 3 . 7 . So lu t ion curves of ejcample 3 .1

28

Example 3.2: In Fig. 3.8, the circuit consists of three nonlinear

resistive elements, two linear resistors and a constant voltage

source. The characteristic of each nonlinear element is given by

Fig. 3.9a and Fig. 3.9b with

h^Cx) =

X - 6 X < -1

5x -10 X > 3

Applying the proposed algorithm, three solutions

(1.200, -0.800, 3.200)^, (1.225,-0.900, 2.725)^, (1.323,-1.290,1.452)-^

are obtained. A solution curve L and all solutions on L are shown

in Fig. 3.10.

hi(x) hoCx)

10

Fig. 3.9a The characteristic Fig. 3.9b The characteristic of a nonlinear element of a nonlinear element

29

CHAPTER IV

SINGULAR JACOBIAN MATRICES

^'1 The Corner Problem

This section is devoted to the study of the corner problem

[13, 16]. When a solution curve reaches a corner, the previous

algorithm cannot determine the next region which is to be entered.

Therefore, the proposed algorithm must be modified accordingly.

A boundary hyperplane in x-space can be represented by

I T ^ x ~ (x|nx = c } where n is the normal vector of the hyperplane

and c is a constant. A corner is a subset of the intersection

of two or more such hyperplanes. Since g = J x + w"*"- y

I T and H = { x | n x = c } , the image of any boundary hyperplane H

of region R is a hyperplane H in the g-space where

H = {g|Jj(i)"'g = c+nTj(^>'' (w^-y )}

Furthermore, the image of a corner in the x-space is a subset

of the intersection of two or more hyperplanes in the g-space.

Thus in the g-space HA.^^g|pg = q}= ^g|^-^ g=c+n J (w -y)}

where p is a nxl vector and q is a scalar, a corner can be

I T represented as a subset o f U = { g | P g = Q } where P is a (nx2)

matrix and Q is a (2x1) vector.

30

31

Let

be a corner on the boundary of region R where P. and Q. are known

Since U is contained in Sp(U . ) , the linear manigold spanned by

U . which is an (n-1) dimensional subspace, there exists a (nxl)

unit vector h^ such that (h^) g =0 for all g U . . Let

i i 1 t>T,b, - - - b , constitute a basis for Sp (U . ) . Then h can 1 2 n-1 gi

be found by

where 0 is an (n-1) dimensional vector.

I T Thus, if a corner represented by U ,= {g| P. g=Q.} which is

on the boundary of region R is hit, then the solution line

segment L. in R must be perturbed such that the point x

on the boundary will just miss the comer (Fig. 4.1). A small

vector V = mh is computed, where m is a small number, such that

g(x ) is perturbed by v. It is always possible to obtain a new

i i point X in R such that

g(x ) = g ( x ) + v and x = x + J v

i' i The point x is then used, instead of x , for the iteration in

R ^

32

After the solution curve moves away from the boundary of the comer

i.e., the region extended has been identified, x is used again

to regain the original solution curve L.

From the above arguments, it is clear that the solution curve

L can be extended into a new region when a corner is hit. Further

more, the piecewise linearity of g implies that no two line

segments in a region possessing a nonsingular Jacobian matrix can

be portions of a solution curve. In the following section, the

problem caused by singular Jacobian matrix and the means for

overcoming this problem will be discussed.

4.2 Singular Jacobian Matrices Problem

An important property which plays a key role in the discussion

of the problem of singular Jacobian matrix is introduced. [l3, 16]

Lemma 4.1: If two regions R and R^ with Jacobian matrix J

and J • respectively are adjacent, i.e., have a common boundary

hyperplane, then the rank of J and J -' differ at most by

one. [l6]

through a region R with nonsingular Jacobian matrix. It is

assumed that L enters a singular region R , as shown in Fig. 4.2

It follows from Lemma 4.1 that J of region R is of rank n-1.

Therefore a nonzero vector d' such that J d'= 0 is uniquely

33

segment L. in R is obtained by

x(X) = x^ + Xd' where 0 <_ X ; X^

and X is determined such that x(X ) = x lies on another

hyperplane H. as indicated in Fig. 4.2 where H„ is between R

and R , By examining the mapping of L. into g-space, it is

found

g(x(X)) = J^^^x(X) + w" = J ' ( x^ + d') + w^

_(i) i , i , i. = J x + w =g(x)

i.e., the whole line segment L. is mapped into the single point

gCx ). Hence all the points on L. satify the Cn-1) equations

r \ n ' ^ o i T T j T i+l) 0 in the g.(x) = 0 , 1 = 1 , 2 , - - - , n-1. If det J

* I 1 * I 1

region of R in which x is lying, then according to Lemma 3.4

and Lemma 3.5, the solution curve can be extended into next region

R^+\ .i+1

In the special case when the next region R has also a

lingular Jacobian matrix, i.e., det J = 0 , then it follows

(i+l) from Lemma 4.1 that the rank of J is either n-1 or n-2

det J^^-^^^ 0

34

since J is of rank n-1. Suppose that J is of rank n-2,

then there exists a 2-dimensional plane S , containing x , such

2 i+1 2 that g(H2ns ) = { g (x )}. Since the hyperplane S H^ containing

X is at least one-dimensional, hence, if x' ? x is a point

9 ' .1.1

on S H^ then g(x') = g(x ). For two linearly independent

vectors (x'- x^) and (x " -x^), J^^\x'-x^) = J^^'^x^'^^-x^)

is satisfied in region R . Thus the rank of J is at most n-2,

which leads to a contradiction. Therefore the rank of J

must be n-1. The solution line segment L. .. can be extended

into region R from x by the same technique stated above,

and the singular Jacobian problem can thus be overcome.

Algorithm A can now be modified so that both the corner

problem and the problem of singular Jacobian problem are taken

care of. The new algorithm is summarized as follows.

Algorithm B:

Step 1. Use Algorithm A to trace the solution curve L.

Step 2. If a comer is hit by the solution curve L, then a

small vector v = mh is computed according to the

perturbation method stated in Section 4.1. A new point

i' i X is then computed and used, instead of x , to iterate

again in region R^. Once the solution curve move away

i+1 from the boundary of the corner, the point x on

the corner is used again to regain the solution curve.

Go to Step 1.

35

Step 3. If a singular region R is identified by the present

solution curve L, then a vector d' is computed such that

J d' = 0 where J is the Jacobian matrix of region

R . Next iterative point x^ is determined by the

direction d'. Go to Step 1.

CHAPTER V

sources are known as as-resitive networks. In this case, the

network equation can be written in the form of

g(x(t), u(t)) = 0 (5.1)

where u(t) is the time varying input vector and x represents

the network variables. Thus, the operating point of the network

is a function of t. One way of solving (5.1) is to solve the

problem at different instants of time, which would be a time-

consuming practice. A continuation algorithm for finding the

solutions (5.1) for all t automatically has been proposed in [22],

Consider the differential equation

^ u(t) = w(t) , u(tQ) = u° (5.2)

where u is assumed to be continously differentiable and its

time derivative, w, is a known function of t. The initial

condition on g is such that s(t-.) = x is a solution of (5.1)

subject to u(tQ) = u . In the x-space, (5.2) is equvalent to

x = -J^^g(x, u) + | ^ w ) , x(tQ) = x° ^^'^^

36

37

where J = r- . Thus, equation (5.3) can be integrated to fine x(t) d X

automatically.

the case where g is a piecewise-linear function. Equation (5.3)

with a piecewise-linear g(x) can now be written as

-1

dt " ^^^^' "' ' au "'' ^^^0 47=-J^^^ (g(x, u) + | 4 w ) , x(tj = x° (5.4)

where J is the Jacobian matrix in the region R in which x

lies. Here it is assumed that J is nonsingular and u(t) is

a small signal vector such that x(t) will remain in R . As it

is noted that since J is a constant matrix, the solution x(t)

f ? can be represented as the superposition of two vectors x' and x

where x' is the solution of g(x, 0) = 0 and x'' is the response

only by the time-varying sources u(t). It is clear that x''(t)

possesses similar waveforms as u(t) since g(x, u) = 0 is a linear

function in region R .

If g(x, u ) = 0 has multiple solutions, then all the solution

points must be used as starting points of (5.4) in order to obtain

a complete family of solution curves. In the following example

a simple circuit is given to illustrate the approach just described,

Example 5.1: Consider the nonlinear resistive circuit in Fig. 5.1

which consists of an ac source in series with a linear resistor

R = 600 ohm and a tunnel diode. The characteristic of the tunnel

38

function.

' 0.006 V - 0.021 V < 4

At t = 0, three operating points are first located by using the

method described in chapter III. The trajectories of v(t) and

i(t) for different Q points are plotted in Fig. 5.2 and Fig. 5.3

respectively.

If u(t) is large enough so that x(t) will not remain in the

region in which x(t-) lies, then the new region that x(t) will

enter must be identified and the Jacobian matrix in that region

should be substituted in (5.4). Proper J • ' s are switched back

and forth in (5.4) for different time intervals and the trajectory

for all t can be obtained accordingly.

600r i(t)

o t

0 .00 4 .00 8 . 0 0 12-00 TIME (SECOND)

16.00 2 0 . 0 0

Fig. 5.2. Output voltages for different starting points"

40

CTiD

:zui"

o o o. 0 .00 ^.OC 2 . 0 0 1 2 . 0 0 16 .00 2 0 . 0 0

T I M E ( S E C O N D ]

Fig. 5.3. Output currents for different startL- g points

41

Multivalued Resistive Nonlinear Networks

Driving point plots and t ransfer cha rac te r i s t i c plots are

important in the analysis of nonlinear r e s i t ive networks because

they can pro-vide a deep insight to the nature of the c i r c u i t .

Bas ica l ly , the two concepts are the same; they are input-output

c h a r a c t e r i s t i c plots showing the relat ionship between a driving

source u. and a cer ta in output variable x . . Several methods 3 1

£18-20,23J are available for finding the input-output plots of

nonlinear r e s i t i v e networks.

Here the technique of continuation scheme for piecewise-linear

analysis of r e s i t i v e nonlinear networks developed in chapter I I I

i s applied to obtain the input-output p lo t s . The network equation,

in general , can be represented by

g(x, u) = 0 (5.6)

where g i s a continous piecewise-linear function, x i s an n-vector

of network va r i ab le s , and u denotes the m-vector of input source.

Since in finding input-output cha rac te r i s t i c plots a l l input

sources except one, say u. , are considered to be fixed, thus

(5.6) i s reduced to

g(x, u.) = 0. (5.7)

Suppose i t i s now required to find the input-output cha rac t e r i s t i c

. - + p lo t X versus u for u 1 u < u , Following the development

42

given in [23], a set of (n+1) equations consisting of (5.7) and

an auxiliary equation g = 0 of the form

g(x, u.) = 0

III, a continuous piecewise-linear curve L can be traced automatically

by the iterative scheme

i"» '' i+1^ X

u

g(x , u.) = 0

where J is the Jacobian matrix in the region of R and x is

the solution point of g(x, u,) = 0. The sign change should be

made when L traverses across a boundary and enters a new region

in which the determinant of the Jacobian matrix has a different

sign,

The advantage of this method is that the input-output plot is

not required to be either input or output controlled, i.e., the

plot may be obtained for multivalued networks. An example is

given in the following to illustrate the approach.

Example 5.2: Consider the tunnel diode problem studied in the

previous example. The input-output characteristic plots, v versus

E and i versus E, is graphed as the input E varies from 0 to 10

volts. The result is shown in Fig. 5.4 and Fig. 5,5.

43

c?

[ N P U T E ( V O L T S ]

Fig . 5 .4 . Multivalued input -output p lo t for the c i r c u i t

.n exaiTDle 5 .2 .

S--00 3 . 0 0 (VOLTS)

10=00

Fig . 5 . 5 . J^ultlvalued input-output p lo t for the. c i r cu i t Ln.esjample. 5.2^

CHAPTER VI

presented for the piecewise-linear analysis of nonlinear resistive

networks. A set of continuous piecewise-linear equations g(x) = 0

is solved by tracing the solution curve L of any n-1 equations of

g(x) = 0 . As a result multiple solutions on L can be obtained.

Furthermore, a sufficient condition for the existence of a unique

simple curve L is given. It is clear that if L is simple, then all

the umltiple solutions of g(x) = 0 can be foimd by the proposed

search scheme.

In Chapter III, the local behavior of L on the boundary is

studied. A theory that guarantees the continuation of the search

algorithm when acrossing the boundary has developed. A method for

finding the starting point also been suggested. Methods for

overcoming the comer problem and singular Jacobian matrix problem

have been presented in Chapter IV. Applications to AC nonlinear

resitive network analysis and driving or transfer characteristic

curve of multivalued nonlinear network are given in Chapter V.

It is found that the algorithm developed here provides an efficient

and systematic way for solving the piecewise-linear nonlinear resis

tive network problems. However, if the solution curve consists of

multiple branches, only the solutions on the branch that contains

45

46

the initial point can be found. A systematic method for locating

a starting point on every branch is worth of further research.

LIST OF REFERENCES

[l] A. F. Malmberg, F. L. Comwell, and F. N. Hofer, "NET-1 network

analysis program", Los Alamos Sciences Lab., Rep. LA-3119, 7090/7094

version, August 1964.

[2] L. D. Milliman, W. A. Massena, and R. H. Dickhaut, "CTRCUS, A

digital computer program for transient analysis of electronic circuit

- user's g\iide", Harry Diamond Labs., The Boeing Co., Seattle, Wash.,

Rep. AD-364-1, January 1967.

[3] H. W. Mathers, S. R. Sedore, and J. R. Sents, "Automated digital

computer program for determining responses of electronic circuits

(SCEPTRE)", vol. 1, IBM Electronic System Center, Owego, N.Y., File

66-928-611, February, 1969.

[4] K. S. Chao, D. K. Liu, and C. T. Pan, "A systematic search method

for obtaining multiple solutions of simultaneous nonlinear equations",

IEEE Trans. Circuits Syst,, vol. CAS-22, pp. 748-753, Sept. 1975.

[5] F. H. Branin, Jr., "Widely convergent method for finding multiple

solutions of simultaneous nonlinear equations", IBM J. Res. Develop.,

vol. 16, no. 5, pp. 506-522, Sept. 1972.

[6] L. 0. Chua and A. Ushida, "A switching-parameter algorithm for

finding multiple solutions of nonlinear resitive circuits". Int. J.

Circuit Theory and Applications, vol. 4, no. 3, pp. 215-239, July 1976.

[7] R. P. Brent, "On the Davidento-Branin method for solving simu

ltaneous nonlinear equations", IBM J. Res. Develop., vol. 16, no. 4,

pp. 434-436, July 1972.

47

ar

48

Lsj E. S. Kuh and I. N. Hajj, "Nonlinear circuit theory: Resistive

networks", Proc. IEEE, vol. 59, pp. 340-355, March 1971.

19J C. T. Pan, "A discrete approach for solving nonlinear equations"

Master's thesis, Texas Tech University, 1974.

ilQj J. Katznelson, "An algorithm for sol-ving nonlinear resistive

networks" Bell System Tech. J. vol. 44, pp. 1605-1620, 1965.

[ll] T. Fujjsawa and E. S. Kuh, "Piecewise-linear theory of nonline

networks", SIAM J. Appl. Math. vol. 22, pp. 307-328, 1972.

[12] T. Ohtsuki, T. Fujjsawa, and E. S. Kuh, "A sparse matrix method

for analysis of piecewise-linear resistive networks", IEEE Trans.

Circuit Theory, CT-19, pp. 571-584, 1972.

[13] M. J. Chien and E. S. Kuh, "Solving piecewise-linear equations

for resistive networks". Int. J. Circuit Theory and Appl., vol. 4,

no. 1, pp. 3-24, January 1976.

[14J M. J. Chien and E. S. Kuh, "Solving nonlinear resistive networks

using piecewi e-linear analysis and simplicial subdivision", IEEE

Trans. Circuits Syst., vol. CAS-24, no. 6, pp. 305-317, June 1977.

[15] L. 0. Chua, "Efficient computer algorithm for piecewise-linear

analysis of resistive nonlinear networks", IEEE Trans. Circuits

Theory, vol. CT-18, pp. 73-85, January 1971.

[I6] T. Ohtsuki, T. Fujjsawa and S. Kumagai, "Existence theorems

and a solution algorithm for piecewise-linear resistive networks",

SIAM J. Appl. Math. 1977.

I17] H. C. So, "On the hybrid description of a linear n-port result

ing from the extraction of arbitrarily specified elements", IEEE Trans

49

[18] L. 0. Chua, Introduction to Nonlinear Network Theory, McGraw-

Hill, New York, N.Y., 1969.

[19J G. H. Meyer, "On Solving nonlinear equations with a one-parameter

operator imbedding", SIAM J. Numer. Analy., vol. 5, pp. 739-752, 1968.

[20] V. C. Prasad and A. Prabhakar, "Input-output plots of piecewise-

linear resistive networks", Proc. 1974 IEEE Int. Symp. Circuits and

Systems, April 1974, pp. 55-59.

12l] R. Saeks and R. A. Decarlo, Interconnected Dynamical Systems,

New York, Marcel Dekker (to appear).

[22] C. T. Pan, "Integration Methods in System Analysis", PH.D.

dissertation, Texas Tech University, 1976.

[23] K. S. Chao and R. Saeks, "Continuation Methods in Circuit

Analysis", Proc. IEEE, vol. 65, no. 8, August 1977.

ELECTRICAL ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirement for

the Degree of

MASTER OF SCIENCE

ACKNOWLEDGEMENTS

I am deeply indebted to Professor Kwong Shu Chao for his

direction of this thesis and to the other members of my committee.

Dr. Donald Gustafson and Dr. A. K. Mitra, for their helpful

criticism. I am also happy to thank my wife and my family for

the constant encouragement throughout this study.

UL

BY A RELATED DIFFERENTIAL EQUATION

2.1 Introduction 4

2.3 Existence and Uniqueness of Solution Curve . 7

III. MULTIPLE SOLUTIONS AND SOLUTION CURVE DEFINED

BY A CONTINUOUS PIECEWISE-LINEAR FUNCTION

3.1 Introduction 9

3.4 Starting Point 21

3.5 Numerical Algorithm 23

JACOBIAN MATRICES

4.2 Singular Jacobian Matrices Problem 32

V. APPLICATIONS

Network 36

of Multivalued Resistive Nonlinear Networks . . . 41

VI. CONCLUSION 45

3.2. The Ebers-Moll Model of a PNP transistor 11

3.3. A solution curve determined by Katznelson's algorithm 13

3.4. A solution curve crosses the boundaries 19

3.5a The circuit of example 3.1 24

3.5b The piecewise-linear approximation of function h for

the tunnel diode 25

3.9a The characteristic of a nonlinear element 28

3.9b The characteristic of a nonlinear element 28

3.10. The solution curve of example 3.2 29

4.1. The solution curve hits a corner 31

4.2. The solution curve enters a singular region 33

5.1. The circuit of example 5.1 38

5.2. Output voltages for different starting podjits 39

5.3. Output currents for different starting points 40

5.4. Multivalued input-output plot for the circuit in

example 5.2 43

example 5.2 44

INTRODUCTION

One of the most basic problems in the area of nonlinear circuit

analysis is the determination of the solutions (equilibrium points)

of nonlinear resistive networks. Research in nonlinear resistive

networks serves as a prerequisite to the understanding of general

nonlinear circuits and systems. This problem is important not only

from a circuit theoretic point of view but even more so from a com

putational point of view. Indeed, an essential part of most nonlinear

transient analysis computer programs is a DC analysis subprogram

[l-3]. In addition, similar problems are encountered in many related

fields such as mathematical economics, flow networks, power systems,

and least-square approximations.

Much work has been done in obtaining multiple solutions of

resistive nonlinear networks [4-9]. Some effort has been devoted

to using the technique of piecewise-linear approximation of a non

linear function [10-16]. For example, in the analysis of a elec

tronic circuit, diode and transistor characteristics can be repre

sented by continuous linear segments. This often gives considerable

insight to the problem and yields quick solutions. In general, by

peicewise-linear approximations, a nonlinear resistive network can

be characterized by the equation

f(x) = y (1.1)

where f(.) is a continuous piecewise-linear mapping of the real

2

n-dimentional Euclidean space R into itself, x is a point in R and

represents a set of chosen variables of a given network and y is an

arbitrary point in R and represents the inputs. Usually, f(.) is

expressed in the form

f(x) = J ' x + w^^^ = y i = 1, 2, , N (1.2)

where J is a constant nxn matrix called the Jacobian matrix and

w is a constant n-vector, both defined in region R . The whole

space R is divided into a finite number N of polyhedral regions

by a finite number of hyperplanes.

Although various techniques have been deveolped for solving

piecewise-linear equations, relatively little has been done in

locating multiple solutions. It is the purpose of this thesis to

develop a systematic search technique for obtaining multiple soluti

ons of (1.1). The continuation method devised by Chao, Liu, and

Pan [4J plays an important role in the scheme developed in this

report.

For convenience and to make the present thesis self-contained,

the continuation method [4] is outlined in Chapter II. In Chapter

III, the general theory of the new algorithm is developed with two

assumptions, namely, all Jacobian matrices may have arbitrary signs

but are not singular and the solution curves never hit a comer.

The dynamic behavior of the search curve is discussed. Approaches

showing how a solution curve can be reached are presented. Formu-

lation of equation (1.1) is also briefly introduced.

In Chapter IV, the corner problem and the problem of singular

Jacobian matrices are considered. Applications to other fields are

discussed in Chapter V. These include AC nonlinear resistive network

analysis and driving point or transfer characteristic curves of mul

tivalued nonlinear resistive networks. Finally, some concluding

remarks are given in Chapter VI.

CHAPTER II

BY A RELATED DIFFERENTIAL EQUATION

2.1 Introduction

g(x) = y (2.1)

or f(x) = g(x) - y = 0 (2.2)

where f or g is a continuous differentia le function from R onto

itself, X and y are both n-dimensional vectors to represent a set

of chosen variables and the inputs respectively.

A systematic search method has been developed by Chao, Liu

and Pan for locating multiple solutions of (2.2). For convenience,

this method is outlined here.

2.2 Iterative Solution Method

Consider a system of differential equations of the form

^ f.(x(t)) = -f,(x(t)) , f,(x(0)) = 0 i = 1, 2, , n-1 dt 1 1 1

where the initial conditions in f. are such that the starting point

X must lie on a space curve L of intersection f.(x) = 0 , i = 1, 2,

- - -, n-1. Since f is not a function of t explicitly, -r— can be dt

written as

dt ax dt "

and i n the x-space (2.3) reduces to a d i f f e r e n t i a l equat ion

X = J""^( -f , , - f , . , - f , , ±f ) ^ , iP ^ x(Q)e L (2.5) i ^ n—i n

where T denotes matrix transpose. Solved by the Zuler intsgraticn

method, equation (2.5) reduces to an iterative scheme

\ ^ l ' \ ^ hJ- x )( -f (x ), - - -, -f .,(V> ^n^V )'

x^ L, k - 1, 2, (2.6)

where h is the step size. The transistion in sign should occur at

the points where the Jacobian J changes sign and the solution points.

The solutions of (2.3) in the f-space are given as

f,(x(t)) = f^(x(0))e'^ = 0 i - 1, 2, , n-1

0 A. These solutions show that for any x * x(0) L, the corresponding

trajectory x(t) from (2,5) remains on L. Theoretically, the signs

of f, for i » l , 2,---,n-l are not important, because the

corresponding trajectory will remain on L so long as f.(x(0)) = 0,

for i = l , 2 , - - - , n-1. But in computational practice, t"r.e f. 1

at each successive iteration may not 'oe precisely zero; the signs

of f. for i = 1, 2-, - - -, n-1 are therefore kept negative to ensure

that the computed trajectory x(t) does not stray too far from the

solution curve L.

Depending on whether the minus or plus sign is being used,

the function f is forced to be attracted to or diverge away n

from zero along L. To ensure the continuation of tracing along

the solution curve L, the sign of f has to be changed at the

solution points of (2.2) and the points on L where the Jacobian

changes sign. The key point lies in a relation between the

determinant of Jacobian J and the directional derivative of f n

in the tangential direction of L

f = ^n = s^(Vf ) (2.8) ^ 3L~

where s is the unit vector in the tangential direction of L

and Vf is the gradient of f (x). The following theorems [4]

determine the conditions of the sign change of f .

Theorem 2.1; If f'(x) is the directional derivative of f (x) n n

in the tangential direction of L at x defined by

f,(x) = 0 , i = 1, 2, n-1 (2.9)

and A., is the (ij)th cofactor of the Jacobian matrix J of

f = (f^, f^, , f^)^, then

f' = det J / I I V I I n

where ||.|| denotes the Euclidean norm and

T ' = ( nl- ^ 2 - 'nr?

Theorem 2.2: The directional derivative of f^(x) in the tangential

direction of L changes sign if and only if the corresponding

Jacobian of f on L changes sign.

Other aspects concerning the convergence of the method and

starting point problem are also detailed in [4j.

2.3 Existence and Uniqueness of Solution Curve

Success in finding all multiple solutions depends hea-vily on

whether (2.9) defines a unique simple curve L. If L is a simple

curve, i.e., a continuously differentiable curve which does not

intersect itself, then a complete tranversal of it enables one to

find all the multiple solutions. On the other hand, if L contains

several branches, then application of the method may lead to those

solutions that lie on the branch containing this starting point only.

The following theorems gives some sufficient conditions for the

existence and uniqueness of the solution curve [9].

Theorem 2.3: Let f be a real function of n real variables and of

class C . If

and (ii) Lim | | f (x) | | = «, as x -»- °°

then any (n-1) equations off. = 0 , i = l , 2 , - - - , n defines a

unique continuously differentiable space curve.

T Theorem 2.4: Let h(x) = ( fj^(x), f2(x), , f^-l^^^ ^ ^® °^

class C . If there exists an (n-1) vector

x_ = ( ^v ^2' - -' \-l' \+l' - -' \ ^^ ^" """ k, 1 <. k < n,

such that

-k (ii) Lim | |h(x) [ | = «, as x , ->• °°

then h(x) = 0 defines a unique simple curve L.

8

Definition: Let U C R • An n x n matrix A(x) is said to be almost

positive definite on U if A(x) is positive definite for all x

in U except at most a set of isolated points on which A(x) is

positive semidefinite.

Theorem 2.5: Let f:R ^R be a C function. If J(x) is almost

positive definite on R , then any (n-1) equations of f. = 0,

i = 1, 2, , n define a unique continuous space curve.

Theorem 2.6: Let f:R->R,n?^2, b e a C map. Let

S = {x^R^I det J(x) = 0} and T = {x € R^ j x e S}

If (i) det J(x) > 0 (or det J(x) < 0 ) for all x^T

and S is at most a set of isolated points,

(ii) limi I f (x) II = °° , as | | x | | - «

then any n-1 equations o f f . = 0 , i = l , 2, - - - , n define

a unique continuous space curve.

CHAPTER III

BY A CONTINUOUS PIECEWISE-LINEAR FUNCTION

3.1 Introduction

In chapter II a systematic search method for solving multiple

solutions of (2.2) is outlined. Considerable computation time is

needed to completely traverse the solution curve, and the success in

finding all the multiple solutions depends heavily on whether or not

the set of equations f.(x)=0, i = l , 2 , - - - , n-1 defines a simple

curve L, i.e., a continuously differentiable curve which does not

intersect itself.

Instead of sol-vrLng a system of nonlinear equations, the piecewise-

linear approximation is obtained for all the nonlinear elements in

a network. As a result, a set of piecewise-linear equations (1.1)

is to be solved. A method due to Katznelson ElO] was originally

applied only to the networks with 2-terminal elements which are

strictly monotonic. Fujjsawa and Kuh [ll] showed that if f is homeo-

morphic then the algorithm due to Katznelson always converges. Here

a continuous function f is said to be homeomorphic from R onto

itself if and only if the equation f(x) = y has a unique solution for

all y.

Kuh, Ohtsuki, Fujjsawa Il23 have shown that as long as all

the Jacobian matrix determinants det J , i = l , 2, - - - - , N

10

in (1.2) have the same sign, there exists at least one solution to

the equation f(x) = y and the Katznelson's algorithm also converges.

This property is refered as the sign condition. Kuh and Chien [l3]

removed the restriction of the sign condition and demonstrated that

if all the Jacobian determinants in the unbounded region have the

same sign, the equation (1.1) has at least one solution. Chien has

also developed an algorithm that the domain of the x-space is divided

into simplices and the beha-vior of the system is approximated by a

linearized system in each simplex [l3]. Chua [l5] also introduced an

iterative scheme for finding multiple solutions. However, the method

is not suitable for networks with a large number of nonlinear elements.

In the following sections, a formulation of (1.2) and a systematic

search method for obtaining its multiple solutions are presented,

where the Jacobian matrices can have abitrary sign. A theoretical

basis for the development of the method is also given.

3.2 Piecewise-linear Approximation

obtaining the piecewise-linear approximation. If the characteristic

of a nonlinear multiport is given by

z = h(x) (3.1)

then h(x) = h(xQ) -H | | | ^ ^ ^ ( X - X Q ) = J ( X Q ) X + w^ 3 ^

where J(x^) = T~| ^ cl w is a constant vector. (J dX I XFX_ U

II

In the case of 2-terminal elements without coupling, this

reduces to the familiar tangent approximation. The Jacobian at

X- denoted by JCX-.) becomes the slope of the nonlinear function

at X-. and w. becomes the vertical axis intercept of the approximate

line segment. If the network contains n nonlinear resistive

elements, each characterized by an n -segment v.

there are N distinct segment combinations where n

i. curve, then 3

Consider a typical resitive nonlinear network as shown symbloicailv

in Fig 3.1 where the n nonlinear elements have been extracted and

are connected across the ports of a linear n-port network. As

indicated by So [17] and Chua [l5]. Fig 3.1 can be completely

characterized by an n x n hybrid matrix A and an n x 1 source vector

u, namely

Ax + u C3.4}

where the components of the n x 1 port vectors 2 and x can be any

combination of port voltage and port current pro-vided that the

corresponding components of 2 and x have the same port number.

N

Linear

Resistive

Network

rig. 3.2 The Toers—Moll model of a ?">!? transistor

12

If each nonlinear resitive element is either voltage controlled of

current controlled, then (3.4) can be written in the form

z(x) = Ax + u (3.5)

T n

z(x) = (z.(x), z„(x), , z (x)) is a nonlinear mapping from R

into R and z.(x) represents the nonlinear characteristics of the

ith nonlinear resistor. In either case, the terminal constraint

imposed by n nonlinear resistors can be expressed by its piecewise-

linear approximation, which is substituted for z in (3.4) or (3.5).

But the piecewise-linear approximation of multiterminal

elements with coupling is much more complicated. Fortunately, most

of coupling elements are of the type that can be expressed as sums

of nonlinear functions V7ith simple variables such as n

z.= E h..(x.) j = 1,2, , n -J 1 = 1 -•

= J^h'..(x.„) + «0 (3.6)

wh ere h'. .(x.^) is the slope of the function h.,(.) at x. ji lO Ji lU.

by ^e = ES< ^ T - » -"r^Cs( ^ - '^

For example, the Ebers-Moll model of a PNP transistor is represented

eb/v^ , r r ^ cb/v^ _ ^

The equivalent circuit model is shown in Fig (3.2) where

^22 = ^CS( ^^"^'"^ - ' ^ ' ^22(\b> ^'-'^

13

Therefore, by specifying the piecewise-linear approximation

to all the nonlinear elements in the circuit, it is clear that

within each region the equation is linear and is of the form

f(x) = J^^^x + w^^^ = y i = 1, 2, ,N (3.10)

where J is a constant nxn matrix, y is the input vector.

The domain of f(.) in R is divided into N polyhedral regions

by a finite number of hyperplanes (or boundaries). A typical

boundary in the x-space can be characterized by the equation

n^x = 0 (3.11)

where n is the normal vector of the hyperplane (or boundary)

and c is a real constant. In (3.10), for example, if x is the

node-to-datum voltage vector and y is the input current source

vector, then J is the node admittance matrix.

3,3 Dynamic Behavior of Solution Curves

Let R , R be continuous regions with a common boundary charac-

14

T

terized by the hyperplane n x = c. Since f is continuous piecewise-

linear, there exists an n-vector r such that

j(l) . j(j) = ,J (3 3)

where J and J ^ are Jacobian matrices of adjacent regions R

and R respectively.

f(x) = J ^ x + w^ = y, i = 1,2, , N

where y is a given input. Fig. 3.3, a continuous curve in the

x-space is determined, based on Katznelson's algorithm

x^(X) = x^ + AJ ^ (y - y= )

where

and X is a scalar parameter, If X = 1, then

i/iN 1 ^ r(l) f T(i) 1 1\

x ( l ) = x + J (y-J x - w )

= J ^ (y-w^).

It is clear that x (1) is a desired solution if it happens to be

in R" . Otherwise, the value of X has to be determined such that

x"''(X) lies on the boundary of R and R . Such a value of X is

denoted by X^. Define x^ = x^(X^) and y" = f(x ), the line

segment connecting x and x is then a portion of the desired

solution curve. We may extend the solution curve beyond x into

R""" in the similar manner.

15

ohtsuki, Kuh and Fujjsawa [l2] showed that if det J and

det J -" have the same sign, the solution curve in the x-space will

indeed enter R . And the solution curve will never enter a region

which has been traced if all the Jacobian matrices determinants

have the same sign. If we remove the condition that all Jacobian

determinants have the same sign, the problem of the continuous traverse

of the complete solution curve becomes much more complicated. Kuh

a d Fujisawa [ll] have derived a equation on the boundary

det j(i>n^j("-l= det A^^'K'"/'^'^ "^ (3.13)

This equation gives a theoretical basis of the development of the

new algorithm presented here.

In what follows, a new algorithm for locating multiple solu

tions of (1.1) is developed. The solution curve is defined and

the local behavior of the solution curve across the boundary is

also investigated. It is assumed that det J ^ 0 and the solution

curve does not hit a comer, but det J may have arbitrary sign.

The comer problem and singular Jacobian problem will be discussed

in next chapter.

Let gCx) = fCx) - y = J ^ x + w^ - y, i = 1,2, , N (3.14)

It is of interest to locate multiple solutions of g(x) = 0. Instead

of applyign Katznelson's algorithm directly to (3.14) starting

from a point selected in one way or another, the iterative scheme

i+1 i,,i..(i) . . i. , iv / i\ . / INNT X = X +X J (-g (x ), -g2(x ), , -gj . Cx ), *g^(x ))

16 x^ € L, i - 0 , 1 , 2 , (3.15)

i s used where the s t a r t i n g po in t x must l i e on a space curve L of

i n t e r s e c t i o n g . (x) = 0 , i = 1,2, , n-1 and X^ > 0 i s the l a r g e s t

p o s i t i v e number such t h a t x and x are in the same reg ion .

0 1 2 As long as X i s chosen to l i e on L, the i t e r a t i o n po in t s x , x ,

~ — remain on L.

Lemma 2.1: If x lies on a space curve L defined by g (x) * 0

1 2 i = 1,2, , n-1, then the successive points x , x , resulted

from (3.15) and the.'line segments joining these points also lie

on L.

Proof: Without lo s ing g e n e r a l i t y , assume x € L i s in region R .

Consider the following d i f f e r e n t i a l equat ion in R

4 r g . ( x ( t ) ) = - g , ( x ( t ) ) , g , (x (0 ) ) = 0, i = 1,2, , n - l d t 1 1 1

4 - 3 ( x ( t ) ) » ± g ^ ( x ( t ) ) , g„(x(0)) » g^^ x(o) = x°c . L (3.16) dt n n n no ,

3y the chain-rule and Euler integration formula,

1..0, 0 ,0,(0)-l, , 0 , , 0 . . 0. , 0. • X (X ) - X + X J^ {~%^{-3i )> -g2(^ )» > "^n-l^*"^ - ' -^n^^ -

(3.17)

where 0 < X < X , X i s the l a r g e s t p o s i t i v e cons tant such t h a t — — m m

x^(X°) i s s t i l l in R^, i . e . x^(X^) l i e s on the boundary of R and m 01

R . The s o l u t i o n of (3.16) in g-space i s given by

g ^ ( x ( t ) ) - g^(x^) e ' ^ » 0, i « 1,2, , a-1

g _ ( x ( t ) ) » g^n^-^. (3.18) o. n 'nO"

This shows t ha t i i x £ L, the corresponding s o l u t i o n cur-ze x ( t ) ,

1 0 i . e . X (A ) in ( 3 . 1 7 ) , remains on L. Following tne same procedure ,

17

in region R , x (X ) is used as the starting point for the next

2 1 iteration. Then it is clear that the line segment joining x (X )

m

and X (X ) remains on L. m

In general, if x = x (X ) is on L, then by the iterative

scheme (3.15), x = x (X ) and the line segment joing x"""

and X also lies on L. By deduction. Lemma 1. is proved.

Let the line segment joining x and x be denoted by L..

Then in fact, L = L^ U L^ U . The minus sign of g for i =

1,2, , n-1 is used for reducing the computation error in each

successive iteration.

Let d^ 4 J^^^"^-g^(xS, , -gn_i(xS, ±g^(xS)^, i = 0,1,2,

(3.19)

Then that the solution curve L passes through each region R ,

i = 0,1,2, in the direction of d where the positive or

negative sign before g (x ) in each region can be determined by n

the following two lemmas. It is assumed that the solution curve

L passes through R in the direction of d then enters region R

at X and traverses R in the direction of d

Lemma 2.2: If there exists no solution in region R and (i) the

determinants of Jacobians J and J have different signs,

then in region R the sign of g should be changed: (ii) if on

the other hand, the determinants of J and J have the same

sign, then in region R the sign of g should not be changed.

18

* * I 1

Proof: Since x and x are on the solution curve L and there * ' * I 1

exists no solution in R , then g (x ) and g (x ) must have the

same sign. Without loss of generality, assume -g (x ) is used

for obtaining d"*" in R . As shown in Fig. 3.4, T 1 T 1+1

n d > 0 and n d > 0 (3.20)

are satisfied on the boundary between R and R where n is

the normal vector of the boundary. Substituting d into (3.20)

(3.19a)

Since x is a point on the solution curve L, (3.19a) becomes

^Tj(i)"l[ 0, 0, , -g^(xS]^ (3.19b)

From the condition of (3.13)

det J n J = det J n J

it is clear that

det J ^ n^J^^^'^O, 0, - -, -g^(x^)]^

- , , i, , , i+lxij .T(i+1) T^ (i+l)"" r n n f i+l^^T = {g (x )/g (x )}detJ' ' J ' L 0, 0, - -, -g^(x )J

(3.13a)

Now, if det J and det J have different signs, then

from (3.13a) n^J^^^ [ 0, 0, , -g^(x^)]^

, T,Ci+l)'''"rn n / i+l^lT and n J^ [0, 0, -—, "gj (.x )J

T i+1 have different signs. To assure n d > 0 on the boundary,

the sign of g should therefore be changed.

19

On the other hand, if det J and det J have the same

sign, then n^J^^^ [0,0, , -g^(x^)]^

and T_(i+1) n J

-1 [0,0, — . -g^(x^"'^]^

will have the same sign and

,i+l ,(i+l)~ r^ f. f i+lxlT d = J L0,0, , -g^ (x )J

T i+1 should be used to assure n d >0 on the boundary. Similarly,

if +g (x ) is used for obtaining d in R the same conclusion result

Lemma 2.3: If there exists a solution in region R , then the

transition in sign of g should occur if det J and det J o °n

have the same sign of g should not occur if det J and det J

have different signs.

Proof: It is clear that g will change sign when the solution

cuirve L passes through a solution point. Hence g (x ) and g (x )

should have different signs.

20

From the same argument as that given in Lemma 2., and

conditions

Tji - T^i _ n d >0, n d >0

.r.A A . ,{1) T Al)-1 . , .(i+l) T ^(i+l)-l and det J n J = det J n J ' ,

Lemma 2.3. is proved immediately.

In view of the above, the solution curve can easily be

taken care of at a local point on the boundary when Jacobian

determinants have arbitrary signs. Therefore the iterative

scheme of (3.15) allows continuation of tracing along the solution

curve L and, as a result, all the solutions on L can be obtained.

If g.(x) = 0,1,2, , n-1 defines a unique simple curve L,

that is, one with only a single branch, the L passes through

all the solutions of (3.14). Once a starting point on L is

obtained, all of the multiple solutions of (1.1) can be located

by using the technique described in this section. The following

lemma gives a sufficient condition for the existence and unique

ness of this solution curve.

Lemma 2.4: Let g(x) be a continuous piecewise-linear mapping of

R^ into itself and let J ^ denote the matrix by deleting the

kth row and nth column of Jacobian matrix J of region R , and

let J , denote the matrix consisting of the first m rows and km

first m columns of the constant Jacobian J, . If there exists

a k, l<JiC^n, such that for each m = 1,2, , n-1, J, , J

21

km • "km ' ""' (N)

J j do not vanish and have the same sign then g.(x) = 0, j = 1,

2, , n-1 defines a unique simple curve.

Proof: Let h(x) = (g^(x), g2(x), , g _;L '' ^

and x_^ = (x^, x^, — , x^.i^x^+i. — , x^)^

If the value of x, is fixed, then the mapping h of R ~ into itself

is still a continuous piecewise-linear mapping. For any x, ,

h(x_ .,u) = \ x_^ + v^(u) - y_^, i = 1,2, , N

where y_^ = (y^.yj, — , yi . , 7^+1. — . y„)^

Since for each m, m = 1,2, , n-1, J, , , J, do not Icm Km

vanish and have the same sign, the mapping h with fixed x, is

a homeomophism of R onto itself. This implies that for any

given input y and any number u of x, there exists one and only

one point T

x_j (u) = (x^(u), X2(u), , X^_;L^^'^' ^+1^^-^' n - ^

such that h(x i,u) = 0. This guarantees the existence of a unique

continuous piecewise-linear function F: R ->R for h(x) = 0 such

that X ,= F(x, ). Hence F(x,) defines a unique simple curve L(x, )

in R for all

A starting point lying anywhere on the intersection L of

g.(x) = 0, i = 1,2, , n-1 has to be found before the root-finding

technique described in the previous section is applied.

22 Let M be a positive number such that for any x » (x., x-,

T , X ) with X, >_ M, X is in an unbounded region. The union of

such unbounded regions is denoted by U. An arbitrary point x in

k k

R is chosen such that R U. With a fixed value of x, = u,

u >.M, a point x = Cx^(u), , Xj^_^(u), u, x^+^(u), , x^(u))^

which s a t i s f i e s g^(x) - 0, i » 1,2, , n-1 i s searched i n the (n-1) dimensional space R

Let g(x) be a continous p i e c e w i s e - l i n e a r mapping of R onto

i t s e l f defined i n ( 3 . 1 4 ) , and l e t J ^ ^ denote the matr ix by

d e l e t i n g the kth row and nth column of Jacobian mat r ix J of

the reg ion R .. In the following a lemma which was proved by

Kuh, Ohtsuki , and Fuijsawa Cl2] i s s t a t e d .

Lemma 2 . 5 . Let g be a continous p i e c e w i s e - l i n e a r mapping from

R i n t o i t s e l f defined by ( 3 . 1 4 ) . For an a r b i t r a r y v, i f det J

i = 1,2, ,N have same s i g n , then the re e x i s t s a t l e a s t one

s o l u t i o n X to the equat ion g(x) = 0. An immediate consequence of

Lemma 2.5 • i s as fo l lows.

Coro l l a ry : I f f o r a given k , 1 _<k<_ n , such t h a t in each region

R- , R" c ,^» ^ ]. <io ^ot vanish and have the same s ign then for

any x, , x, >^M, there e x i s t s a t l e a s t one s o l u t i o n for

g^(x) - 0 , i « 1, 2 , , n - 1 .

23

This corollary gives a sufficient condition for the existence

of starting point x . The convergence to x from an arbitrary

initial guess in R-', R-'c U is guaranteed by the Katznelson's

algorithm.

in the following steps.

Algorithm A:

Step 1. Find an initial point x in an unbounded region R according

to section 3.4. Set j = 0.

Step 2. Compute g(x ).

Step 3. If j=f 0, compute d according to Lemma 2. and Lemma 3.

^ 1- . ^0 .(O)"-'-, , 0, , 0.,T Otherwise, compute d = J ( -g,(x ) , - - - , -g (x )) .

i i i * Step 4. Compute x = x* + d . If x is in R then x = x is a

solution.

Step 5. Compute x-' = x" + X~ d , where X-" > 0 is the maximum value • • * * *

such that x(X) = x-"' + Xd" , 0 <_ X <_ X^ is in R^. If X^- » go

to step 7.

Step 6. Otherwise, identify region R-" . Set j = j +1 and go to

step 2.

Step 7. Set j = 0. Repeat step 2. to step 6. except in step 3. that

if j = 0 compute d° = J^°^ ( -g^{^^) . . +g^(x°))'^

gand in step 5. that if X-' ->•=», stop.

Two examples are given here for illustration,

24

Example 3.1. Consider the nonlinear circuit of Fig. 3.5a, consisting

of two tunnel diodes and two constant current sources and a linear

resistor. The piecewise-linear approximation of function h for the

tunnel diode is given in Fig. 3.5b with

' 2x X < 3

1.5x-6.5 x > 5

8^^\x) J ^ ^ ^ + w^ - y 0, i - 1, 2, , 9

are obtained. All regions are shown in Fig. 3.6. The equation

associated with each region is given as:

g^»Cx) 3

25

1= 2] +

-6 .5 '

-6 .5

In this example, y, i.e. the input current source is given by

y =

13.5

13.5

h(x)i

h for the tunnel diode

26

R3

R2

R1

R^

R5

R*

R '

R3

R^

X

Fig. 3.6. The regions in the x-space

If the solution curve L of g,»0 is traced, then the points

on L with g^» 0 are the desired solutions. For x.. >, 5, in each

7 8 9 unfaoxjnded region R , R , R oo"* 2.5 has the same sign and does

not vanish. Hence from Lemma 5, for a given value of x. = 5.5 > 5

0 T a corresponding solution x » Cy'5, 6.25) is found by solving

7 8 9 0

g_. Cxi = 0 in the unbounded regions R , R , R . Using x as a

starting point, the search along the curve L of g =0 leads to

five solutions C5.714, 5.714)^, (6.316, 4.211)^, (7.154, 2.115)*,

(4.211, 6.316)^, C2.115, 7..154) .

In fact, a simple solution curve does exist in this example

even though Lemma 4, is not satified. Similarly, if g O is

traced, the same five solutions are obtained. The solution curves

and all solutions are shown in Fig. 3.7.

27

IG.OO

F i g . 3 . 7 . So lu t ion curves of ejcample 3 .1

28

Example 3.2: In Fig. 3.8, the circuit consists of three nonlinear

resistive elements, two linear resistors and a constant voltage

source. The characteristic of each nonlinear element is given by

Fig. 3.9a and Fig. 3.9b with

h^Cx) =

X - 6 X < -1

5x -10 X > 3

Applying the proposed algorithm, three solutions

(1.200, -0.800, 3.200)^, (1.225,-0.900, 2.725)^, (1.323,-1.290,1.452)-^

are obtained. A solution curve L and all solutions on L are shown

in Fig. 3.10.

hi(x) hoCx)

10

Fig. 3.9a The characteristic Fig. 3.9b The characteristic of a nonlinear element of a nonlinear element

29

CHAPTER IV

SINGULAR JACOBIAN MATRICES

^'1 The Corner Problem

This section is devoted to the study of the corner problem

[13, 16]. When a solution curve reaches a corner, the previous

algorithm cannot determine the next region which is to be entered.

Therefore, the proposed algorithm must be modified accordingly.

A boundary hyperplane in x-space can be represented by

I T ^ x ~ (x|nx = c } where n is the normal vector of the hyperplane

and c is a constant. A corner is a subset of the intersection

of two or more such hyperplanes. Since g = J x + w"*"- y

I T and H = { x | n x = c } , the image of any boundary hyperplane H

of region R is a hyperplane H in the g-space where

H = {g|Jj(i)"'g = c+nTj(^>'' (w^-y )}

Furthermore, the image of a corner in the x-space is a subset

of the intersection of two or more hyperplanes in the g-space.

Thus in the g-space HA.^^g|pg = q}= ^g|^-^ g=c+n J (w -y)}

where p is a nxl vector and q is a scalar, a corner can be

I T represented as a subset o f U = { g | P g = Q } where P is a (nx2)

matrix and Q is a (2x1) vector.

30

31

Let

be a corner on the boundary of region R where P. and Q. are known

Since U is contained in Sp(U . ) , the linear manigold spanned by

U . which is an (n-1) dimensional subspace, there exists a (nxl)

unit vector h^ such that (h^) g =0 for all g U . . Let

i i 1 t>T,b, - - - b , constitute a basis for Sp (U . ) . Then h can 1 2 n-1 gi

be found by

where 0 is an (n-1) dimensional vector.

I T Thus, if a corner represented by U ,= {g| P. g=Q.} which is

on the boundary of region R is hit, then the solution line

segment L. in R must be perturbed such that the point x

on the boundary will just miss the comer (Fig. 4.1). A small

vector V = mh is computed, where m is a small number, such that

g(x ) is perturbed by v. It is always possible to obtain a new

i i point X in R such that

g(x ) = g ( x ) + v and x = x + J v

i' i The point x is then used, instead of x , for the iteration in

R ^

32

After the solution curve moves away from the boundary of the comer

i.e., the region extended has been identified, x is used again

to regain the original solution curve L.

From the above arguments, it is clear that the solution curve

L can be extended into a new region when a corner is hit. Further

more, the piecewise linearity of g implies that no two line

segments in a region possessing a nonsingular Jacobian matrix can

be portions of a solution curve. In the following section, the

problem caused by singular Jacobian matrix and the means for

overcoming this problem will be discussed.

4.2 Singular Jacobian Matrices Problem

An important property which plays a key role in the discussion

of the problem of singular Jacobian matrix is introduced. [l3, 16]

Lemma 4.1: If two regions R and R^ with Jacobian matrix J

and J • respectively are adjacent, i.e., have a common boundary

hyperplane, then the rank of J and J -' differ at most by

one. [l6]

through a region R with nonsingular Jacobian matrix. It is

assumed that L enters a singular region R , as shown in Fig. 4.2

It follows from Lemma 4.1 that J of region R is of rank n-1.

Therefore a nonzero vector d' such that J d'= 0 is uniquely

33

segment L. in R is obtained by

x(X) = x^ + Xd' where 0 <_ X ; X^

and X is determined such that x(X ) = x lies on another

hyperplane H. as indicated in Fig. 4.2 where H„ is between R

and R , By examining the mapping of L. into g-space, it is

found

g(x(X)) = J^^^x(X) + w" = J ' ( x^ + d') + w^

_(i) i , i , i. = J x + w =g(x)

i.e., the whole line segment L. is mapped into the single point

gCx ). Hence all the points on L. satify the Cn-1) equations

r \ n ' ^ o i T T j T i+l) 0 in the g.(x) = 0 , 1 = 1 , 2 , - - - , n-1. If det J

* I 1 * I 1

region of R in which x is lying, then according to Lemma 3.4

and Lemma 3.5, the solution curve can be extended into next region

R^+\ .i+1

In the special case when the next region R has also a

lingular Jacobian matrix, i.e., det J = 0 , then it follows

(i+l) from Lemma 4.1 that the rank of J is either n-1 or n-2

det J^^-^^^ 0

34

since J is of rank n-1. Suppose that J is of rank n-2,

then there exists a 2-dimensional plane S , containing x , such

2 i+1 2 that g(H2ns ) = { g (x )}. Since the hyperplane S H^ containing

X is at least one-dimensional, hence, if x' ? x is a point

9 ' .1.1

on S H^ then g(x') = g(x ). For two linearly independent

vectors (x'- x^) and (x " -x^), J^^\x'-x^) = J^^'^x^'^^-x^)

is satisfied in region R . Thus the rank of J is at most n-2,

which leads to a contradiction. Therefore the rank of J

must be n-1. The solution line segment L. .. can be extended

into region R from x by the same technique stated above,

and the singular Jacobian problem can thus be overcome.

Algorithm A can now be modified so that both the corner

problem and the problem of singular Jacobian problem are taken

care of. The new algorithm is summarized as follows.

Algorithm B:

Step 1. Use Algorithm A to trace the solution curve L.

Step 2. If a comer is hit by the solution curve L, then a

small vector v = mh is computed according to the

perturbation method stated in Section 4.1. A new point

i' i X is then computed and used, instead of x , to iterate

again in region R^. Once the solution curve move away

i+1 from the boundary of the corner, the point x on

the corner is used again to regain the solution curve.

Go to Step 1.

35

Step 3. If a singular region R is identified by the present

solution curve L, then a vector d' is computed such that

J d' = 0 where J is the Jacobian matrix of region

R . Next iterative point x^ is determined by the

direction d'. Go to Step 1.

CHAPTER V

sources are known as as-resitive networks. In this case, the

network equation can be written in the form of

g(x(t), u(t)) = 0 (5.1)

where u(t) is the time varying input vector and x represents

the network variables. Thus, the operating point of the network

is a function of t. One way of solving (5.1) is to solve the

problem at different instants of time, which would be a time-

consuming practice. A continuation algorithm for finding the

solutions (5.1) for all t automatically has been proposed in [22],

Consider the differential equation

^ u(t) = w(t) , u(tQ) = u° (5.2)

where u is assumed to be continously differentiable and its

time derivative, w, is a known function of t. The initial

condition on g is such that s(t-.) = x is a solution of (5.1)

subject to u(tQ) = u . In the x-space, (5.2) is equvalent to

x = -J^^g(x, u) + | ^ w ) , x(tQ) = x° ^^'^^

36

37

where J = r- . Thus, equation (5.3) can be integrated to fine x(t) d X

automatically.

the case where g is a piecewise-linear function. Equation (5.3)

with a piecewise-linear g(x) can now be written as

-1

dt " ^^^^' "' ' au "'' ^^^0 47=-J^^^ (g(x, u) + | 4 w ) , x(tj = x° (5.4)

where J is the Jacobian matrix in the region R in which x

lies. Here it is assumed that J is nonsingular and u(t) is

a small signal vector such that x(t) will remain in R . As it

is noted that since J is a constant matrix, the solution x(t)

f ? can be represented as the superposition of two vectors x' and x

where x' is the solution of g(x, 0) = 0 and x'' is the response

only by the time-varying sources u(t). It is clear that x''(t)

possesses similar waveforms as u(t) since g(x, u) = 0 is a linear

function in region R .

If g(x, u ) = 0 has multiple solutions, then all the solution

points must be used as starting points of (5.4) in order to obtain

a complete family of solution curves. In the following example

a simple circuit is given to illustrate the approach just described,

Example 5.1: Consider the nonlinear resistive circuit in Fig. 5.1

which consists of an ac source in series with a linear resistor

R = 600 ohm and a tunnel diode. The characteristic of the tunnel

38

function.

' 0.006 V - 0.021 V < 4

At t = 0, three operating points are first located by using the

method described in chapter III. The trajectories of v(t) and

i(t) for different Q points are plotted in Fig. 5.2 and Fig. 5.3

respectively.

If u(t) is large enough so that x(t) will not remain in the

region in which x(t-) lies, then the new region that x(t) will

enter must be identified and the Jacobian matrix in that region

should be substituted in (5.4). Proper J • ' s are switched back

and forth in (5.4) for different time intervals and the trajectory

for all t can be obtained accordingly.

600r i(t)

o t

0 .00 4 .00 8 . 0 0 12-00 TIME (SECOND)

16.00 2 0 . 0 0

Fig. 5.2. Output voltages for different starting points"

40

CTiD

:zui"

o o o. 0 .00 ^.OC 2 . 0 0 1 2 . 0 0 16 .00 2 0 . 0 0

T I M E ( S E C O N D ]

Fig. 5.3. Output currents for different startL- g points

41

Multivalued Resistive Nonlinear Networks

Driving point plots and t ransfer cha rac te r i s t i c plots are

important in the analysis of nonlinear r e s i t ive networks because

they can pro-vide a deep insight to the nature of the c i r c u i t .

Bas ica l ly , the two concepts are the same; they are input-output

c h a r a c t e r i s t i c plots showing the relat ionship between a driving

source u. and a cer ta in output variable x . . Several methods 3 1

£18-20,23J are available for finding the input-output plots of

nonlinear r e s i t i v e networks.

Here the technique of continuation scheme for piecewise-linear

analysis of r e s i t i v e nonlinear networks developed in chapter I I I

i s applied to obtain the input-output p lo t s . The network equation,

in general , can be represented by

g(x, u) = 0 (5.6)

where g i s a continous piecewise-linear function, x i s an n-vector

of network va r i ab le s , and u denotes the m-vector of input source.

Since in finding input-output cha rac te r i s t i c plots a l l input

sources except one, say u. , are considered to be fixed, thus

(5.6) i s reduced to

g(x, u.) = 0. (5.7)

Suppose i t i s now required to find the input-output cha rac t e r i s t i c

. - + p lo t X versus u for u 1 u < u , Following the development

42

given in [23], a set of (n+1) equations consisting of (5.7) and

an auxiliary equation g = 0 of the form

g(x, u.) = 0

III, a continuous piecewise-linear curve L can be traced automatically

by the iterative scheme

i"» '' i+1^ X

u

g(x , u.) = 0

where J is the Jacobian matrix in the region of R and x is

the solution point of g(x, u,) = 0. The sign change should be

made when L traverses across a boundary and enters a new region

in which the determinant of the Jacobian matrix has a different

sign,

The advantage of this method is that the input-output plot is

not required to be either input or output controlled, i.e., the

plot may be obtained for multivalued networks. An example is

given in the following to illustrate the approach.

Example 5.2: Consider the tunnel diode problem studied in the

previous example. The input-output characteristic plots, v versus

E and i versus E, is graphed as the input E varies from 0 to 10

volts. The result is shown in Fig. 5.4 and Fig. 5,5.

43

c?

[ N P U T E ( V O L T S ]

Fig . 5 .4 . Multivalued input -output p lo t for the c i r c u i t

.n exaiTDle 5 .2 .

S--00 3 . 0 0 (VOLTS)

10=00

Fig . 5 . 5 . J^ultlvalued input-output p lo t for the. c i r cu i t Ln.esjample. 5.2^

CHAPTER VI

presented for the piecewise-linear analysis of nonlinear resistive

networks. A set of continuous piecewise-linear equations g(x) = 0

is solved by tracing the solution curve L of any n-1 equations of

g(x) = 0 . As a result multiple solutions on L can be obtained.

Furthermore, a sufficient condition for the existence of a unique

simple curve L is given. It is clear that if L is simple, then all

the umltiple solutions of g(x) = 0 can be foimd by the proposed

search scheme.

In Chapter III, the local behavior of L on the boundary is

studied. A theory that guarantees the continuation of the search

algorithm when acrossing the boundary has developed. A method for

finding the starting point also been suggested. Methods for

overcoming the comer problem and singular Jacobian matrix problem

have been presented in Chapter IV. Applications to AC nonlinear

resitive network analysis and driving or transfer characteristic

curve of multivalued nonlinear network are given in Chapter V.

It is found that the algorithm developed here provides an efficient

and systematic way for solving the piecewise-linear nonlinear resis

tive network problems. However, if the solution curve consists of

multiple branches, only the solutions on the branch that contains

45

46

the initial point can be found. A systematic method for locating

a starting point on every branch is worth of further research.

LIST OF REFERENCES

[l] A. F. Malmberg, F. L. Comwell, and F. N. Hofer, "NET-1 network

analysis program", Los Alamos Sciences Lab., Rep. LA-3119, 7090/7094

version, August 1964.

[2] L. D. Milliman, W. A. Massena, and R. H. Dickhaut, "CTRCUS, A

digital computer program for transient analysis of electronic circuit

- user's g\iide", Harry Diamond Labs., The Boeing Co., Seattle, Wash.,

Rep. AD-364-1, January 1967.

[3] H. W. Mathers, S. R. Sedore, and J. R. Sents, "Automated digital

computer program for determining responses of electronic circuits

(SCEPTRE)", vol. 1, IBM Electronic System Center, Owego, N.Y., File

66-928-611, February, 1969.

[4] K. S. Chao, D. K. Liu, and C. T. Pan, "A systematic search method

for obtaining multiple solutions of simultaneous nonlinear equations",

IEEE Trans. Circuits Syst,, vol. CAS-22, pp. 748-753, Sept. 1975.

[5] F. H. Branin, Jr., "Widely convergent method for finding multiple

solutions of simultaneous nonlinear equations", IBM J. Res. Develop.,

vol. 16, no. 5, pp. 506-522, Sept. 1972.

[6] L. 0. Chua and A. Ushida, "A switching-parameter algorithm for

finding multiple solutions of nonlinear resitive circuits". Int. J.

Circuit Theory and Applications, vol. 4, no. 3, pp. 215-239, July 1976.

[7] R. P. Brent, "On the Davidento-Branin method for solving simu

ltaneous nonlinear equations", IBM J. Res. Develop., vol. 16, no. 4,

pp. 434-436, July 1972.

47

ar

48

Lsj E. S. Kuh and I. N. Hajj, "Nonlinear circuit theory: Resistive

networks", Proc. IEEE, vol. 59, pp. 340-355, March 1971.

19J C. T. Pan, "A discrete approach for solving nonlinear equations"

Master's thesis, Texas Tech University, 1974.

ilQj J. Katznelson, "An algorithm for sol-ving nonlinear resistive

networks" Bell System Tech. J. vol. 44, pp. 1605-1620, 1965.

[ll] T. Fujjsawa and E. S. Kuh, "Piecewise-linear theory of nonline

networks", SIAM J. Appl. Math. vol. 22, pp. 307-328, 1972.

[12] T. Ohtsuki, T. Fujjsawa, and E. S. Kuh, "A sparse matrix method

for analysis of piecewise-linear resistive networks", IEEE Trans.

Circuit Theory, CT-19, pp. 571-584, 1972.

[13] M. J. Chien and E. S. Kuh, "Solving piecewise-linear equations

for resistive networks". Int. J. Circuit Theory and Appl., vol. 4,

no. 1, pp. 3-24, January 1976.

[14J M. J. Chien and E. S. Kuh, "Solving nonlinear resistive networks

using piecewi e-linear analysis and simplicial subdivision", IEEE

Trans. Circuits Syst., vol. CAS-24, no. 6, pp. 305-317, June 1977.

[15] L. 0. Chua, "Efficient computer algorithm for piecewise-linear

analysis of resistive nonlinear networks", IEEE Trans. Circuits

Theory, vol. CT-18, pp. 73-85, January 1971.

[I6] T. Ohtsuki, T. Fujjsawa and S. Kumagai, "Existence theorems

and a solution algorithm for piecewise-linear resistive networks",

SIAM J. Appl. Math. 1977.

I17] H. C. So, "On the hybrid description of a linear n-port result

ing from the extraction of arbitrarily specified elements", IEEE Trans

49

[18] L. 0. Chua, Introduction to Nonlinear Network Theory, McGraw-

Hill, New York, N.Y., 1969.

[19J G. H. Meyer, "On Solving nonlinear equations with a one-parameter

operator imbedding", SIAM J. Numer. Analy., vol. 5, pp. 739-752, 1968.

[20] V. C. Prasad and A. Prabhakar, "Input-output plots of piecewise-

linear resistive networks", Proc. 1974 IEEE Int. Symp. Circuits and

Systems, April 1974, pp. 55-59.

12l] R. Saeks and R. A. Decarlo, Interconnected Dynamical Systems,

New York, Marcel Dekker (to appear).

[22] C. T. Pan, "Integration Methods in System Analysis", PH.D.

dissertation, Texas Tech University, 1976.

[23] K. S. Chao and R. Saeks, "Continuation Methods in Circuit

Analysis", Proc. IEEE, vol. 65, no. 8, August 1977.

Recommended