The Schwarz-Christoffel Transformation and Polynomial Root Clustering1

THE SCHW ARZ-CHRISTOFFEL TRANSFORMATION AND POLYNOMIAL ROOT CLUSTERINGl T. A. Bickart* and E. I. Jury** ·Department of Electn·cal and Computer Engineen"ng, Syracuse University, Syracuse, New York 13210, U.S.A . •• Department of Electn·cal Engz"neen"ng and Computer Sciences and the Electronics Research Laboratory, University of Caltfornia, Berkeley, Calzfornia 94720, U. S. A.

Abstract.

A sequence of tests on derived polynomials to be strictly Hurwitz polynomials is shown to be equivalent to a given (typically real) polynomial having all its zeros in an open sector, symmetric with respect to the real axis, in the left half-plane. The number of tests needed is at most 2 + [(In k)/(ln 3»), where k is the integer associated with the central angle n/k of the sector. An extension of this result on the sector as a region of root clustering is given which shows that only a limited number of tests are needed to verify that the roots are clustered in a region composed as the intersection of a set of primative (sector-like) regions. The results reported evolve from invokation of a collection of mappings on the complex plane defined by a particular collection of Schwarz-Christoffel transformations.

Ke ywords.

Polynomials; root clustering criteria; Hurwitz criterion; stability criteria; Schwarz-Christoffe1 transformation. INTRODUCTION

Let us now suppose that the characteristic frequencies are the roots (or zeros) of the polynomial P in the variable s. The property we are considering--re1ative stability of degree a--can be placed in an abstract framework. This is accomplished through the

If the characteristic frequencies of a lumped system are confined to an open sector symmetric with respect to the real axis in the left half-plane, such as depicted in Fig. 1, then the system is said to be relatively stable (Jury, 1974, p.23; Leonhard, 1952; Stojic and Si1jak, 1965). The degree of relative stability is a, where 2a is the central angle of the sector.

DEFINITION: A polynomial P is said to be an R[a]-polynomial if for some a E (O,n/2] aU the zeros of P a.-»e in the sector S = {s:

a

Polynomial root clustering in a sector has been considered by others (Anderson, Bose, and Jury, 1974; Bose and Jury, 1973; Davidson and Ramesh, 1970; Jury, 1974, pp. 23-27; Marden, 1966, pp. 191-193; Stojic and Si1jak, 1965). The research results have been tests for root clustering in a sector. The result reported herein simplifies and extends the test devised by Bose and Jury using a SchwarzChristoffel transformation (Bose and Jury, 1973). Specifically, we shall describe a test for R[a]-stabi1ity when a = rr/2k (k = 1,2, ... ) requiring

Im s

Fig. 1

larg{-s}l
Sector in left half-plane.

N

1

(k

1)

(la)

N

2

(k

2)

(lb)

N

1 + [(In k)/(ln 3») (k

1

Research sponsored in part by a special grant from the Syracuse University Senate Research Committee to the first author and by the National Science Foundation under Grant ENGR76-2l816 to the second author.

N

n 3 ; n

=

1,2, .•• )

(lc)

2 + [(In k)/(ln 3)] (k

1171

= >

n 3 and k -; 3 ; n .. 1,2, ... ) (ld)

1172

T. A. Bickart and E. I. Jury

2

tests on a polynomial to be strictly Hurwitz.

Ims PRINCIPAL RESULT The Schwarz-Christoffel transformation (Silverman, 1972, pp. 332-339). s

= -(-p)

l/m

(2)

Lffi'lfnlfft.fHtt----

(3)

m= I

Re s

maps each of the sectors Iarg { -se <

7T /2m }

j2 7TR. /m} I

( R. = 0, ••• ,m-I)

onto the open left half-plane

lm s

~=

{p: Re p < DJ.

(4)

These sectors for m = 1, .•• ,4 are depicted in Fig. 2 . Let PR. denote the polynomial PR. (s) = P(se

j27T

R. /m)

( R. = O, .•• ,m-l)

Then, rather obviously, all the zeros of P

= Po

are in S 7T /2m

= SO/2 if 7T mR.

and only if

all the zeros of PR. are in S 7T /2m'

m=2

Now set (5)

Im s

and (6)

m

Note: S 7T /2m is the preimage by the trans-

hn~'Ifn~~---

Res

formation (2) of~. Clearly, all the zeros of P are in S 7T /2m only if all the zeros of m m Pare inS 7T /2rn' Unfortunately, the converse relation is not true. However a reciprocal relation does exist, as stated in Let

THEOREM 1:

Z

O}

(7)

K = [(In k)/(ln 3)].

(8)

m

m=3

= is:

prn(s)

and

Then P is an R[7T / 2k ]-pol.ynomial. if and onl.y if 1\

rn

wher e

"""k 1\

{ z

c

m

Srn }, 7T /2rn

(9)

denotes the l.ogical. 'and ' operation

m=4

and (k

= 1)

(lOa) Fig. 2

Sectors

.5 7TR. / 2rn ( R. =0, ••. ,rn-l;

m=l,2,3,4). 2

The notation [r], r a real number, is used to signify the integer part of r.

The Schwarz-Christoffel Transformation

{1,2}

K

{1,3,9 ... 3 } u {k}

[only if]

PROOF:

s=

{s:

pes)

(k = 2)

(lOb)

(k ~ 3).

(lOc)

and 2 and the fact that N of (1) is the cardinaltiy of the set ~k' P can be shown

Let

= OL

(11)

Suppose P is an R[n/2k]-polynomial; that is, suppose 'it c S n /2k' Then clearly 'Zc Sn /2m for all m € .l't • But, as previously observed k Zc S /2 only i f Z c Sm/ 2 • Therefore (9) n

m

m

m

m

Z c S:/2m'

then

n

[if] Obviously, i f 2

must be true.

1173

cS~/2 ' "m

Hence, by (9), we know that

to be or not to be an R[n/2k]-polynomial by carrying out N tests for a polynomial to be a strictly Hurwitz polynomial. This procedure is presented as the P is an R[n/2k]polynomial if and only if, for all

R[n/2k]-STABILITY TEST:

mE. .)(k> nomial.

cf1

is a strictly Hurwitz poly-

As an illustration of this result consider the polynomial pes)

=

s3 + 7s

2

+ l6s + 10.

(12) Now, as can be easily shown m

(13)

nmli.)(k Sn/2m = Sn/2k'

To determine whether it is an R[n/8]-polynomial (k = 4) we must show that each of the following three polynomials is a strictly Hurwitz polynomial: Ql(p)

It therefore follows that ~c Sn/2k; equivalently, P is an R[n/2k]-polynomial.

•

s

=

(15)

Is m

Im

1

=

degree {Qm(p)}.

[only if] As (2) maps S:/2m onto :t and as

Xi c S:/2m' it follows that the zeros of Qm(p) are in ~ __ Qm(p) is a strictly Hurwitz polynomial. [if] As the preimage of any point of ~ by (2) is in S:/2m and as the m c

.5 n / 2m •

As an immediate consequence of Theorems 1

then for an affirmative test pes) will have al1 its zeros not in Sn/2k but in Sn/2k truncated on the right at s = -a, as depicted in Fig. 3a. If the replacement is by

Ql(p -a) and Ql(_p _ S) are strictly Hurwitz polynomials with 0 < a < S. then for an affirmative test the zeros of pes) wil1 be in S n/2k truncated on the right

zeros of Qm(p) are inX, it follows that

Zm

=1

Q (p - a) is a strictly Hurwitz polynomial with 0 < a (17)

degree{P(s)}

P

is replaced by

s

Note:

+ 1036s 3 + 1000\ 3

Ql(p) is a strictly Hurwitz polynomial

applied to pm yields a polynomial

in p.

6

If the constituent test corresponding to m

PROOF: Observe, from the manner by which is defined in (5) that the transformation

= pm(s)

+ 37s

SECONDARY RESULTS

p

is a strictly Hurwitz polynomial .

pm[_(_p)l/m]

9

By the Lienard-Chipart criterion (Jury, 1974, pp. 22-23), each of these polynomials is strictly Hurwitz. Therefore, pes) is an R[n/8]-stable polynomial.

(16)

m+l (-1)

=

p

p3 + 57p2 + 10056p + 100000

The statement

pm(s)

s

_s12 + 57s 8 - 10056s 4 + 10000\ 4 s =-p

is true if and only if Qm(p)

+ l6s + 10\

s

The needed criterion is to be found in

c S:/2m

2

3 P + 37p2 + 1036p + 1000

(14)

~m

s3 + 7s

3 p + 7p2 + l6p + 10

We must now consider the problem of validating each statement of (9); that is, of validating

THEOREM 2:

=

•

at s = -a and on the left at s = -S, as depicted in Fig. 3b. Both of these results are easily verified.

1174

T. A. Bi cka rt and E. I. Jury

Im 5

Im 5

(a)

(a)

Im

Im

5

5

.....

~--~~~~~/~_~8--Res

(b) Fig. 3

(b)

Truncated sectors.

Fig. 4

By applying the R[n/2k]-stability test to the polynomial pes - 0 ) for some real 0 , an affirmative test establishes the region of root clustering to be .5 n /2k transI"ated to the left by 0 , as depicted in Fig. 4a. When the constituent test for m = 1 is also replaced as discussed above, with 0 ~ a < S , the region is that depicted in Fig. 4b. In the preceding paragraph we indicated how, by simple alteration of the R[n/2k]-stability test and/or by a translation of the spectrum of the polynomial, the region of root clustering could be made a truncated and/or translated sector. Much more can be be achieved. We shall first provide a sense of this through several examples. If it can be shown that Ql(p - a ) with 0

<

3

a and Q (-p) are strictly Hurwitz polynomials, then the roots of pes) are confined to a pair of sectors symmetric with respect to the real axis and truncated on the right, as depicted in Fig. Sa. On the other hand, if, corresponding to the polynomial pes - 0), it can be shown that 1

Q (p - a ) with 0

~

a,

QlC_p - S) with a < S , and Q4(_p) are strictly

Translated sectors.

Hurwitz polynomials, then the roots of pes) are confined to the pair of translated sectors symmetric with respect to the real axis and truncated on both the left and right, as illustrated in Fig. Sb. As a last example, 1 3 suppose it can be shown that Q (p), Q (p), 4 and Q (-p) are strictly Hurwitz polynomials, then the roots of pes) are confined to the pair of sectors depicted in Fig. Sc. The variety of achievable regions is exceedingly great as the following general result ( R. )

makes clear. Let P (s) = P( ~R. s + vR. ). ( R. m) th . Suppose Q ' is the m polynom1al derived from p( R. ) as in (5).

Then the zeros of P

are in .$( R. ,m,n) = {s: s =_~ [_( ~ p+ v )]l/m+ R. n n and pc;.~ } i f and only i f Q( R. ,m,n)(p)

v

t R. ,m)(~nP

Q

+ v ) is a strictly Hurwitz polyn nomial in p. This result is a simple extension of Theorem 2--the new feature is the linear transformations of the s-plane before and of the p-plane after invoking the Schwarz-Christoffel transformation. The counterpart of the R[ n /2k]-stability test-Theorem 2 with Theorem l--is: The zeros of

The Schwarz-Christoffel Transformation

, m , n)

Il t

(1 , 1 , 1)

1

(t

Im s

~ ~

J. .....

Re s

-a . . . .

1175

(1 , 3

,

v

t

Il n

0

1

1)

1

0

(1 , 1 , 1)

1

(1 , 3 , 1)

1

-B -B

v

n 0

1

0

-1

0

-1

0

Then, the roots of P are in the diamond shaped region depicted in Fig. 6a. As a small variation on this case, suppose QCt,m,n) is a strictly Hurwitz polynomial for each (t,m,n) E ~ , where with B > a > 0

(a) Ct

Im s

(1 , 1

J. --'---+----=__~I-:-t---

, m , n)

Re s

,

Il t

v

t

Il n

v

1

0

1)

1

0

3 , 1)

1

0

1

0

,

1)

1

-B

-1

0

(1 , 3 , 1)

1

0

-1

-a

(1

,

(1 , 1

n

3

Then, the roots of P are in the not diamond shaped region depicted in Fig. 6b.

Ims / /

( b)

Im s -*"f+NYH'H+.,.",f----

-/3

....I....--.L.-:.--::Pt----

Res

Re s ( a)

Im s

(c) Fig. 5

Pairs of sectors.

¥ • n S( t ,m,n) if and only P a.e 1n (t,m,n)E~ t if Q( ,m,n) is a strictly Hurwitz polynomial for all (t,m,n) e: ~ , ~ being a suitable set of triplets of index values. The number of constituent tests for a polyno@ial to be a strictly Hurwitz polynomial is the card inality of~. To further illustrate this rather general conclusion, we cite two cases.

Res

(b) Fig. 6

Regions of root clustering.

( t m n)

Suppose Q " is a strictly Hurwitz polynomial for each ( t ,m , n) e ~ , where with B> 0

There is a further level of generalization; however, we will do no more than point it out. In the extension of Theorem 2 we pointed out above that the new teature was linear transformations before and after the Schwarz-Christoffel transformation. These linear transformations of the s-plane and p-plane can be replaced by a more general

T. A. Bi ckar t and E. I. Jury

1176

bilinear (or Mrdbius) transformation (Hille, 1959, pp. 46-58). CONCLUDING DISCUSSION Though we have only shown regions which are symmetric with respect to the real axis, as would be appropriate when P is a real polynomial, the regions need not exhibit such symmetry. For almost all values of ~ ~ , v ~, ~ n'

and vn as complex constants the regions,

in fact, will not be symmetric with respect to the real axis. Regions located selectively throughout the complex plane are appropriate to the treatment of root clustering of a complex polynomial, which of course, P may be. The primary constraint of the method of composing a region for root clustering developed herein would appear to be that the central angles of the sectors which comprise each region S:/2m--such regions being, in a sense, the basic ingredients for the more general regions--are restricted to be divided by a positive integer. However, it is possible for one to show that this can be overcome by proper composition of the general region u

(~,m,n) EJ)

S U ,m,n) .

We began this report by focussing on sectors because such regions are easily categorized. This rather general extension we have described--we trust in a convincing manner, but without formal proof--is not particularly easy to apply as there are no composition rules for an arbitrary region, nor any reason to believe that an arbitrary region can be composed as U( "D,m,n ) E ___ n S ( ~ ,m,n). Continued research on root clustering will focus on restrictions which reduce the general region to a categorizable set of regions. We have not considered a Schwarz-Christoffel transformation more general than that of (2) because it is not clear how to carry out a root clustering test on the resulting polynomial of an algebraic function--P(s) with s an algebraic function of p other than (2). Herein, of course, we solved that problem by creating the polynomial Qm(p) for which we had only to carry out a test for it to be a strictly Hurwitz polynomial. As a pleasant side result the degree of Qm was just that o f P. In another paper we examine the root clustering problem by composing the complement of the region as a union of disks and halfplanes (Bickart and Jury, 1978). That point of view is the dual of that followed here-composition of the region as an intersection of simpler regions.

ACKNOWLEDGEMENT We herewith express our appreciation to Professor Nirmal K. Bose (University of Pittsburgh) for his reading of our preliminary version of this paper. His comments and suggestions aided us in the preparation of this final version of our paper. REFERENCES Anderson, B.D.O., N.K. Bose, and E.I. Jury (1974). A simple test for zeros of a complex polynomial in a sector. IEEE Trans . Automatic Control ., AC-1 9 , pp. 437-438. Bickart, T.A., and E.I. Jury (1978). Regions of polynomial root clustering. J . of Franklin Inst ., to appear. Bose, N.K., and E.I. Jury (1973). The Schwarz-Christoffel transformation applied to stability problems. Proc .

Asilomar Conf. on Cir cuits and Systems , pp. 148-152. Davison, E.J., and N. Ramesh (1970). A note on the eigenvalues of a real matrix.

IEEE Trans . Automatic Contr ol ., AC-1 5 , pp. 252-253. (1959). Analytic Function Theory , Vol . 1, Ginn, Boston, MA.

Hille, E.

(1974). Inners and Stability of Dynamic Systems , Wiley, New York.

Jury, E.I.

Leonhard, A. (1952). Relative damping as a criterion for stability and an aid in finding the roots of a Hurwitz polynomial. Automatic and Manual Control ., Butterworth, London. Marden, M. (1966). Geometry of Polynomials , American Mathematical Society, Providence, RI. (1972). Introductory Complex Analysis , Dover, New York.

Silverman, R.A.

Stojic, M.R., and D.D. Siljak (1965). Generalization of Hurwitz, Nyquist, and Mikhailov stability criteria.

IEEE Trans . Automatic Contr ol, AC-1 0 , pp. 250-255.