"Cascaded network optimization program," IEEE Trans. Microwave ...

300

IEEE

TRANSACTIONS

ON MICROWAVE

Math. Its App7., voL 10, PP. 39+403, 1972. 13rodie, A. R. Gourlay, and J. Greenstad}, “Rank-one and rank-two corrections to positive definite matrmes expressed ~o}roduct form,” J. inst. Math. Its Appt., vol. 11, pp. 7>82,

K. W.

-“.

[50]

-.

J. W. Bandler, J. R. Popovi& and V. K. Jha, “Cascaded network optimization program,” this issue, pp. 300-308. [37] John W. Bandler and Berj L. Bardakiian, “Least pth optimization of recursive digital filters,” IEEE Trans. Audio Electroacoust., vol. AU-21, pp. 460470, Ott. 1973. [381 and T. V. Srinivasan. “Gra. . J. W. Bandler. N. D. Markettos. client minimax techniques for ‘system modeling,” ~nt. J. L’@t. Sci., vol. 4, pp. 317-331, 1973. [39] M. R. Osb?rne.and G. A. Watson, “An algorithm for minimax

[36]

[51]

[52] [53]

[54]

aPProxlmatIOn m the non-linear case,” Comput. J., Vo]. U, pp. 63-68, 1969. [40] J. W. Bandler and P. A. Macdonald, “OpttiIzation of microwave networks by razor search,” IEEE Trans. Microwave Theor~ Tech., vol. MTT-17, pp. 552–562, Aug. 1969. [41] J. W. Bandler,. T. V. Srinivasan, and C. Charalambous, “Mlnimax optimization of networks bv grazer search,” IEEE Trans. Microwave Theory Tech., vol. MTT-20, pp. 59&604, Sept. 1972.

[42] L. S. Lasdon and A. D. Waren, “Optimal Trans. bounded, 10SSYelements,” IEEE [43]

[44] [45] [46]

[47]

[48]

159–174. 1973. [56] A: ‘R. C&nj

Theory,

~; “Cornpu~ati?nal strained muumlzatlon Management Sci., vol.

1972. -----

JOHN

Absfracf—A will

variant or

timization Presently,

The

and

networks program

deletions

of

is organized

a variety

of two-port

C-type

sections

as fixed

or variable

circuit

[60] [61]

[62]

upper

Optimization

in such

elements lumped

and

and lower

time-in-

and

a way that

all-pass

future

addi-

constraints,

are readily

D-type

is presented

liiear

filters

specifications,

and all-pass

between

package

cascaded

as microwave

performance

and

program

certain

such

methods,

all-pass

[59]

MEMBER, IEEE, JADRANKA

computer

optimize

and T. Pletrzykowski, “A penalty function method converging directly to a constrained optimum,” Dep. Combinatorics and Optimization, Univ. Waterloo, Waterloo, Ont., Canada, Rep. 7>11 ., ----1072. conjugate direction method to M. J. Best, ‘[FCD: : A feasible solve linearly constrained optimization problems,” Dep. Combinatorics and Optimization, Univ. Waterloo, Waterloo, Ont., Canada, Rep. CORR 72-6, 1972.. M. E. Mokari-Bolhaesan and T. N. Trick, “Computer-aided design of distributed-lumped-active networks,” IEEE Trans. Circuit Theory, vol. CT-18, pp. 187–190, Jan. 1971. R. W. Newcomb, Active Integrated Czrcwit Synthesis. Englewood Cliffs, N. J.: Prentice-Hall, 1968. John W. Bandler and Rudolph E. Seviora, “Current trends in network optimization,” IEEE Trans. Microwave Theory Tech., vol. MTT-18. .,. un. . 1159–1 170. Dec. 1970. N. D. Markettos, “Optimum system ‘modelling using recent M. Eng. dissertation, McMaster Univ., gradient methods,” Hamilton, Ont., Canada, Apr. 1972.

[58]

algorithm for the sequential uncontechnique for nonlinear programming,” 10, pp. 601-617, 1964.

BANDLER,

user-oriented

analyze electrical

networks. tions

W,

op-

implemented.

dk.tributed

elements,

sections

can be treated

bounds

on the parame-

Program

R. POPOVI~,

ters.

Adjoint

AND

network

or Fletcher

optimize

least

in the

objective cretion, the

in the

package, d@al

way

(if

general,

is written

insertion

any).

response

The

IV,

The

can be called

to

Charalambous

an

at the

user’s

dis-

10SS, group

delay,

and

program

is

specifications

overlapping

in Fortran

incorporated.

and

simultaneously,

coefficient,

in which in

are

JHA

methods

of Bandler

incorporating

of,

which

formulas

sense

constraints

number

K.

optimization

pth

reflection

parameter

flexible any

function input

VIRENllRA

sensitivity

Fletcher-Powell

6400 Manuscript received July 23, 1973; revised November 9, 1973. This work was supported by the National Research Council of Canada under Grant A7239 and by the Communications Research Laboratory of McMaster University,. J. W. Bandler and J. R. Popovlc are with the Communications Research Laboratory and Department of Electrical Engineering, McMaster University, Hamilton, Ont., Canada. V. K. Jha was with McMaster University. He is now with RCA Limited, Ste.-Anne-de-Bellevue, P. Q., Canada.

optimization using nondifferentiable J. Num. Anal., vol. 10, pp. 760–784,

[57] A. R. Corm

vol.

Network

“Constrained function,” SIAM

penalty

CT-13, pp. 175-187, June 1966. R. Klessig and E. Polak, “A method of feasible directions using function approximations, with applications to minimax problems,” J. Math. Anal. Appl., vol. 41, pp. 583-602, 1973. L. S. Lasdon, Optimization Theory for Large Systems. New York: Macmillan, 1970. W. I. Zangwill, Nonlinear Programming: A Unijkd Approach. Emrlewood Cliffs. N. J.: Prentice-Hall. 1969. A. ‘D. Waren, L. S. Lasdon, and D. F. Suchman, “Optimization in engineering design,” P?’oc. IEEE, VO1. 55, pp. 1885–1897, NOV. 1967. A. V. Fiacco and G. P. McCormick, “The sequential unconstrained minimization technique for nonlinear programming; a primal-dual method,” Management S’ci., vol. 10, pp. 360-366, 1WA

Cascaded

that

[55]

design of filters with Circuit

VOL. MTT-22, NO. 3, MARCH1974

AND TECHNIQUES,

R. Fletcher, “A class of methods for nonlinear programming with termination and convergence properties,” in Integer and Nonlinear Programming, J. Abadie, Ed., Amsterdam, The Netherlands: North-Holland, 1970. — “An exact penalty function for nonlinear programming with’inequalities,” Math. Progra., vol. 5, pp. 129–150, 1973. J. Kowalik. M. R. Osborne. and D. M. Rvan. “A new method for constrained optimization problems,” O$er.’ Res., vol. 17, pp-~ 973–WR 196!2. . . T. P&~z~kowski, “An exact potential method for constrained maxima,” MAM J. Num. Anal., vol. 6, pp. 299–304, 1969. A. v. Flacco and G. P. McCormick, “Extensions of SUMT for nonhnear programming: Equahty constraints and extrapolw tion,” Management i%., vol. 12, pp. 816-829, 1966. R. R. Allan and S. E. J. Johnsen, “An algorithm for solving nonlinear programming problems subject to nonlinear inequality constraints,” Comput. J., vol. 13, pp. 171–177, 1970. M. Avriel, “Solution of certain nonlinear programs involving Theory Appl., vol. 11, pp. r-convex functions, ” J. Optimiz.

[49]

J. Inst.

[35]

TliEORY

are handled

frequency has been

particularly

tested

bands.

at The

on a CDC

computer.

I. INTRODUCTION

A

USER-ORIENTED computer program package is presented that will analyze and optimize certain cascaded linear time-invariant networks such as microwave filters and all-pass networks

in the frequency

domain.

d d:

BANDLER

CASCADED

State-of-the-art functions [l]

techniques

of many

and Fletcher

The

adjoint

ciriuit largely

approximation

ciency

program

additions

and

evaluation

is organized

[3]–[5] is in least pth

design

are incorprogram

advance

similar

in such a way

of performance

for

computer

a significant

over previous

or deletions

ith

*i

in @i-

work

[7].

that

future

specifications,

and

distributed

lines

including

resonant

elements open

and antiresonant

such

and

as lossless

shorted

stubs,

con-

circuits,

transmission

all-pass

C-type

sections, and all-pass D-type sections can be handled. Upper and lower bounds on all relevant parameters can be specified

by’ the user. At the user’s

discretion,

a least

pth objective function or a sequence of least pth objective functions incorporating simultaneously input reflection coefficient, insertion loss, relative group delay, and parameter constraints (if any) are automatically created. Finite values of p greater than 1 can be used. It is felt that

the

which

program

response

is particularly

specifications

flexible

in the

are handled

way

in

‘at any number

of, in general, overlapping frequency bands. The package, which is written in Fortran

IV,

Some of the upper

II.

as well

problem For which straint

and lower

is specified notational may

bounds

approximation

sought

in the ‘[best”

results

which

as follows.

or lower

if .si is an

1 +1.0

=

weight upper

Then number

the problem

essentially

of inequalities

becomes one of satisfying

where all subscripts are dropped to avoid F, or C. y will be called the approximating understood

that

(4) must include

and constraints

a

(4)

Z(?J-S) 0,

4

or con-

x; such that

(3)

all e,

Cj(+) C.j Clj

bound

s;,

– 1.0 if .s~is a lower bound.

~

(*,)

way.

a specification

response

and a corresponding

the package A point

design inequalities

s,,

and feasi-

where

problem

in general,

vari-

be the same

meaningful

we define

eewz(y–s)

The discrete

may

An acceptable

in a physically

simplicity

be an upper

bound,

xi

as

THEORY

solves can be stated,

independent

ble design is one for which the inequalities are satisfied. It is the job of the designer to ensure that his design

The Probkm

which

of the

(single specification/constraint).

has been

tested on a CDC 6400 digital computer. Some of the many test examples will be presented here to illustrate the approach. Examples of input and output actual execution times will be given.

point

Some of the upper bounds may be + co and’some of the lower bounds may be – m, in which case they are ignored.

straints, optimization methods, and circuit elements are readily implemented. Presently, a variety of two-port lumped elements, including resistors, inductors, and capacitors as well as lumped

sample

able $.

and Charalambous

as titer

work

it represent

of

to the user.

of gradient

by Bandler

present

and organization

The

areavailable

method

for such tasks The

with

minimization

in the frequency domain State-of-the-art techniques

and generalized associated

gradient

301

PROGRAM

such as the Fletcher-Powell

[2]methods

developed

[6].

OPTIMIZATION

in

variables

network

elements employed.

porated

NETWORK

as to the values

of p, the weighting factors w and the artificial margin ~. Discussion of these parameters is available in the literature [5], [6], [8]–[10] and so will not be repeated here. An important point to remember, however, is that the first optimization with a particular value of p will determine whether the specifications and constraints can be satisfied for any other value [9].

302

IEEE

Performance are clearly objective

specifications

treated

parameter

in essentially

function.

Fig.

the least pth

objective

straints,

Fig.

and

and

1(a)

the

1(b)

when a single upper

eter is desired

(see also Charalambous

To

Translation

and Artificial

distinguish

sponses

the different

ways

are calculated

we have

the of

for

the

various

particularly

employed

ber of response functions

TECHNIQUES,

1974

MARCH

‘O”s:’”’

“O\ \u n,zw then y ~

L

C, Vy +-- VC

if

CODE 2

A

H==l

(11) I (r–

1)2. ~ z’ < rzuforanyr

then y h

< {1,2,...,n,)

7“”

IMPLEMENTATION

The Subprograms Fig. 2 shows a block diagram of the subprograms comprising the network optimization program. A brief description of these subprograms is given in this section. CANOPT ( CAscaded Network Optimization program) is the subroutine called by the user. It reads and organizes input

data,

determines

subprograms,

and prints

user to enter,

z’ as in out

(10),

results.

conveniently,

single

1“

OBJECT

CANOPT

F., Vy +- vF,. III.

controls It

(5) without function (5)

also enables

specifications

the

(upper

l+=

E

COOE C

COOE D

Fig.

Table

the other

1:

I to

show

2.

the

The

subprograms.

circuit

presently

elements

incor-

porated. The Circuit

Configuration

equals lower) by setting the parameter x to O. The program splits these into the upper and lower specifications which it is designed to handle. Subroutine OBJECT computes the objective function (5) and the gradients (6). Calculation of e as in (7) is performed through function subprogram ERROX. Subroutine APPROX is responsible for calculating y and Vy as in (11).

The package will optimize a cascade connection of the two-port elements listed in Tables II and III. Elements 1–15 may be connected in any order (sequentially from the source to the load) using as many as required or as many as the computer being used can accommodate. The first six elements are one-parameter lumped ele-

OPTIM1 performs

ments.

optimization

by

and 0PTIM2 by the Fletcher–Powell for a summary grammed

of the features

and the parameters

the user. Tables

11 and III

the

Fletcher

method.

and options which

expand

must

method

See Table currently

I

pro-

be specified

some of the items

by of

the

Their

parameter

user to his center

values frequency

should and

be normalized source

appropriately, as outlined in the Appendix. The next four elements are three-parameter cuits.

They

are characterized

by resonant

by

resistance, tuned

cir-

or antiresonant

et al.:

BANDLER

CASCADED

NETTV0R3C

PROGRAM

OpTII.dIZATtOI.J

303 TABLE

SUMMARY

OF FEATURES,

I

OPTIONS,

AND

PARAMETERS

REQUIRED

— Features

Type

Objective

Least Ptb

Functions

Parameters

Options

1