- Email: [email protected]
Polynomial Root Clustering
Polynomial Root Clusteringt by
T. A. BICKART
of Electricul
Department Syracuse, and
and
Computer
Engineering,
Syracuse
University,
New York 13210
E. I. JURY
Department Electronics
of Electrical Engineering and Computer Research Laboratory, University of California,
Sciences Berkeley,
and
the
California
94720
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. 7’he number of tests needed is at most 1 + [(In k)/(ln 3)1, where k is the integer associated with the central angle r/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 application of a collection of mappings on the complex plane defined by a particular collection of Schwarz-Christoflel transformations. ABSTRACT:
I. Introduction 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 [(6) p. 23, (7), (lo)]. The degree of relative stability is (Y,where 2cx is the central angle of the
sector. 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 consideringrelative stability of degree a-can be placed in an abstract framework. This is accomplished using Definition : A polynomial P is said to be an R[a]-polynomial if for some CYE (0, m/2] ull rhe zeros of P are in the sector .Yw= {s: /arg {-.~}I <(Y}. Polynomial root clustering in a sector has been considered by others [(l), (3), (4), (6) pp. 23-27, (8) pp. 191-193, (lo)]. 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 Schwarz-Christoffel transformation (3). Specifically, we shall describe a test for R[a]-stability when a = r/2k
t Research sponsored in part by a special grant from the Syracuse CJniversity Senate Research Committee to the first author and by the National Science Foundation under Grant ENG76-21816 to the second author.
0 The Franklin Institute001~032/79/1101~87$02.00/0
487
7’. A. Bickart
and E. 1. Jury
Re s
FIG. 1. Sector in left half-plane.
(k = 1,2,...)
requiring N = 1+ [(ln k)/(ln 3)]
tests on a polynomial
to be strictly
(1)
Hurwitz.?
II. Principal Result The Schwarz-Christoffel
transformation
[(9) pp. 332-3391
s = -(-p)“” maps
each of the sectors %,?n
onto
(2)
the open
= {s: larg {-sei*W”m>j < ~/2m)
(I = 0, . . . , m - 1)
(3)
left half-plane Z’={(p:
Rep
(4)
These sectors for m = 1 , . . . ,4 are depicted in Fig. 2. Let PI denote the polynomial P,(s) = P(seiz?rum) (l= O,...,m - 1). Then, rather obviously, all the zeros of P = PO are in YW,2m= SPzjlZ,,,if, and only if, all the zeros of Pt are in P’&,,. Now set P” = rI;‘,-,‘P(
(5)
and x%n
= u i%1%2m.
(6)
Note : Y:Qm is the preimage by the transformation (2) of 2. Clearly all the zeros of P are in Ymlzm only if all the zeros of P” are in Y’J$,,. Unfortunately t The notation [rj. r a real number, is used to signify the least integer greater than or equal to r. Alternatively, [rJ is used to signify the greatest integer less than or equal to r.
488
Journal ofTheFranklin Institute Pergamon PressLtd.
Polynomial Root Clustering Im
5
Im s
Re s
Re s
m-2
m=l Im
s
Im s
Re s
Re s
m=/l
m=3
FIG. 2. Sectors Lf?&,,,(e = 0,. . . , m - 1; m = 1,2,3,4).
the converse stated in
relation is not true. However,
a reciprocal
relation does exist, as
Theorem 1. Let z& ={s: P”(s) = 0)
(7)
K = [(ln k)/(ln 3)J.
(8)
and Then P is an R[r/2k]-polynomial
if, and only if,
A m._MkEn = xbII> where A denotes the logical “and” operation A& ={l,
(9)
and
3,9 ,..., 3K}U{k}.
(10)
Proof: [only if] Let e={S:P(s)=O}. Suppose P is an 2 = yn/&n for all Y”_,*,,,. Therefore [if] Obviously
(11)
R[r/2k]-polynomial; that is, suppose 5%t Y’m,Zk.Then clearly m E .I%,. But, as previously observed, 3 c YmIZ,,,only if %, c (9) must be true. if Z, c .P’&,,, then ZEc .YFlzm. Hence, by (9), we know that 3 = l-l mE”uA(t~%n.
Vol. 308, No. 5, November 1979 Printed in Northern Ireland
(12) 489
T. A. Bickart and E. I. Jury Now, as can be easily shown,
n rn&uk xkrn=elk.
It therefore
follows that % t .Y,,,2k;equivalently,
(13)
P is an R[?r/2k]-polynomial.
0 Suppose K and k were not related as in (8). In that case it would be easy to show, for all non-negative integers K, that (13) would be valid when 3K 5 k I 3 K+* and would not be valid when 3Ki1 < k. It therefore follows with K defined as in (8) that, as a function of k, the rate of increase of the cardinality of JU, is as small as possible. We must now consider the problem of validating each statement of (9); that is, of validating %n = x&m (m ~JUThe needed criterion is to be found in Theorem II.
(14)
Theorem II. The statement %?, c XLn
(1%
Q”(p) = P”(s) Js-+Iyp
(16)
is true if, and only if,
is a strictly Hurwitz polynomial. Proof: Observe, from the manner by which Pk is defined in (5) that the transformation (2) applied to P” yields a polynomial in p: P”[-(-p)““‘]
= P’“(s)),~~=(_~)~+L~ = Q”(p).
(17)
Note: degree {P(s)} =degree {Q”‘(p)}. [only $1 As (2) maps Y&,, onto 2 and as 2 c LY~,~,,,it follows that the zeros of Q”(p) are in Z-Q”(p) is a strictly Hurwitz polynomial. [if] As the preimage of any point of 2 by (2) is in 0 x&m and as the zeros of Q”‘(p) are in 2, it follows that Z,,, c .Y$,,,. As an immediate consequence of Theorems I and II and the fact that N of (1) is the cardinality of the set .&, P can be shown to be or not to be an R[r/2k]-polynomial by carrying out N tests for a polynomial to be a strictly Hurwitz polynomial. This procedure is presented as the R[rr/2k]-stability test: P is an R[m/2k]-polynomial if, and only if, for all m E A,,
Q” is a strictly Hurwitz polynomial.
As an illustration of this result, consider the polynomial P(s)=s3+7sz+16s+10. To determine wheather it is an R[?r/8]-polynomial (k = 4) we must show that each of the following three polynomials is a strictly Hurwitz pOlynOmial: Q'(p)=s3+7~2+16s+10),=,=p3+7p2+16p+10, Q3(p)=s9+
37s"+1036s3+1000),+,=p3+37p2-+1036p+1000,
Q4(p)=-~1Z+57s8-10056s4+10000~,~=_,, =p3+.57p2+10056p+100000. 490
Polynomial Root Clustering
By the Lienard-Chipart criterion [(6) pp. 22-231, each of these polynomials is strictly Hurwitz. Therefore P(s) is an R[r/8]-polynomial. III. Secondary Results If the constituent
test corresponding
to m = 1:
Q’(p) is a strictly Hurwitz polynomial, is replaced by: Q ‘(p - a) is a strictly Hurwitz polynomial with 0 I a, then for an affirmative test, P(s) will have all its zeros not in .YPrrlZk but in .Ym,2k truncated on the right at s =-a, as depicted in Fig. 3(a). If the replacement is by: Ql(p -a)
and Q’(-p -b) are strictly Hurwitz polynomials with 05 a 5 b,
then for an affirmative test, the zeros of P(s) will be in 5p?r/2ktruncated on the right at s = -a and on the left at s = -b, as depicted in Fig. 3(b). Both of these results are easily verified. By applying the R[r/2k]-stability test to the polynomial P(s-d) for some real d, an affirmative test establishes the region of root clustering to be Ym,2k translated to the left by d, as depicted in Fig. 4(a). When the constituent test for m = 1 is also replaced as discussed above, with d 5 a I b, the region is that depicted in Fig. 4(b).
Re s
Re s
ia)
ib) FIG. 3. Truncated
sectors.
Re s
Re s
(a1
(b)
FIG. 4. Translated Vol. 308, No. 5, November 1979 Printedin Northern Ireland
sectors.
491
T. A. Bickart and E. I. Jury In the preceding paragraph we indicated how, by simple alteration of the R[m/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. Even more can be achieved as we next illustrate through several examples. IV. Examples If it can be shown that Q’(p -a) with Osa and a’(--p) are strictly Hurwitz polynomials, then the roots of P(s) are confined to a pair of sectors symmetric with respect to the real axis and truncated on the right, as depicted in Fig. 5(a). On the other hand, if, corresponding to the polynomial P(s -d), it can be shown that Q’(p - a) with d CCa, a’(--p - b) with a < b, and Q”(-p) are strictly Hurwitz polynomials, then the roots of P(s) 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. 5(b). As a last example, suppose it can be shown that Q’(p), Q”(p), and a”(-p) are strictly Hurwitz polynomials, then the roots of P(s) are confined to the pair of sectors depicted in Fig. 5(c). The variety of achievable regions is exceedingly great as the following general result makes clear: Let PC” (s) = P(qs + /3,); suppose Q(l*m)is the m-th = polynomial derived from PC’) as in (5). Then the zeros of P are in Sp(‘Vm7n) (s: (s - &)“’ + (- q)“’ (~,p + A,,) = 0 and p E .Z’e)if, and only if, Q(l,m.“)(p) = Q(*,“‘) ( K~P+ A,) is a strictly Hurwitz polynomial in p. This result is a simple extension of Theorem II-the new features are the linear transformations of the s-plane
Re s
Re s
I
(a) Im s
Re s
FIG. 5. Pairs of sectors. 492
(,b)
Polynomial Root CZustering
before and of the p-plane after invoking the Schwarz-Christoffel transformation. The counterpart of the R[v/2k]-stability test (Theorem II with Theorem I) is: The zeros of P are in n(l,m,njod Y (‘Jw) if, and only if, Q(‘rm*“)is a strictly Hun&z polynomial for all (l,m,n)Ed, d being a suitable set of triplets of index values. The number of constituent tests for a polynomial to be a strictly Hurwitz polynominal is the cardinality of ~4. To further illustrate this rather general conclusion, we cite two cases. Suppose Q(l,m*“)is a strictly Hurwitz polymonial for each (l,m,n)E.d, where with b 50
(6 m, n) aI 6
d
K,
A,
(l,l,l)
1
0
1 0
(1,3,1>
1
0
1 0
(2,1,2)
1 -b
-1
0
(2,3,2)
1 -b
-1
0.
Then the roots of P are in the diamond shaped region depicted in Fig. 6(a). As a small variation on this case, suppose Q(l.“‘+) is a strictly Hurwitz polynominal for each (l,m,n) E Se, where with b > a > 0 (1, m, n) (~1 PI (1,L
se
4
A,
1) 1
0
1
0
(1,3,1)
1
0
1
0
(2,1,2)
1 -b
(1,3,3)
1
-1
0
0 -1 -a3.
Then the roots of P are in the not-diamond shaped region depicted in Fig. 6(b). There is a further level of generalization which subsumes the former. Therefore, we will adopt similar notation in briefly describing it. Let q denote the degree of P. Set P(I) (s) = (ns + 6,)” P[(qs + f$)/(y,s + SJ], where q and 7/l are such that q/n is not a zero of P. (It then follows that the degree of PC’) is also 77.)Suppose Q(‘,“‘) is the m-th polynomial derived from P(l) as in (5). Set
Im s
Ims
Res
Re s
(a)
(b)
FIG.6. Regions of root clustering. Vol. 308, No. 5, Novemkr 1979 Printed in Northern Ireland
493
Q(h,n)
(p) = (p,p + u,)” Q’**“’[(~,p + h,)/(&p + v,,)], where K,, and p,,, are such that K,,/&, is not a zero of Q ‘Lm:. Now the zeros of P are in Lt’((“‘,“)= {s:(&s-fir)‘” (p+,p+~,,)+(yrs-a[)” (~,p+h,)=O and p~2’} if, and only if, Q(P,m,n)is a strictly Hurwitz polynomial. This simple extension of Theorem II incorporates bilinear (or Mobius) transformations [(S) pp. 46-581 of the s-plane before and of the p-plane after invoking the Schwarz-Christoffel transformation. (The degree preserving conditions on q and -yl and on K, and CL, guarantee that each (finite) zero is transformed into a (finite) zero.) As above, the R[rr/2k]-stability test counterpart is: The zeros of P are in if, and only if, Q(‘.m*n)1s . a strictly Hurwitz polynomial for all (I,m,n) E d. To illustrate this very general result, we offer the following two examples: Suppose Q”,“‘+) is a strictly Hurwitz polynomial for each (I,m,n) E d, where with a > 0
C/f, m*“) n(Lm.nk~
d
(1, Wr) o[ (l,l, 1) -a
Pr ?‘I 8, ‘G A, P, % 0 1 -1 1 0 0 1
(1,3,1)-a
0
1 -1
1 0
0
1.
Then the roots of P are in the tear-drop shaped region depicted in Fig. 7(a). Next suppose Q(r+.“) is a strictly Hurwitz polynomial for each (f,m,n) E Se, where with a > 0
(l,l,l) sQ
(2,3,1)-a
10
0 0
11
1 -1
0
0
1
1 0
0
1.
Then the roots of P are in the not-simply-connected 7(b).
region depicted in Fig.
(b)
FIG. 7. Regions of root clustering.
494
Polynomial Root Clustering
V. 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 (Ye,PI, -yl,and 8, and of K,, A,, CL,, and v,, 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 YF,,-,,,- 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 n,,,,,,,,, Z@“‘+). We began this paper by focussing on sectors because such regions are easily categorized. This quite 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(I,m,n)Ed .9”f3m+).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 cause, we solved that problem by creating the polynomial am(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 Q” was just that of P. In another paper we examine the root clustering problem by composing the complement of the region as a union of disks and half-planes (2). 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 (1) B. D. 0. Anderson, N. K. Bose and E. I. Jury, “A simple test for zeros of a complex polynomial in a sector”, IEEE Trans. Autom. Control. AC-19, pp. 437-438, 1974. (2) T. A. Bickart and E. I. Jury, “Regions of polynomial root clustering”, J. Franklin Inst., Vol. 304, pp. 149-160, 1977. Vol. 308, No. 5. November 1979 F’rintedin Northern Ireland
495
T. A. Bickart
and E. I. Jury
(3)N. K. Bose and E. I. Jury, “The
Schwarz-Christoffel transformation applied to stability problems”, Proc. Asilomar Conf. on Circuits and Systems, pp. 148-152, 1973. (4) E. J. Davison and N. Ramesh, “A note on the eigenvalues of a real matrix”, IEEE Trans. Autom. Control., AC-15, pp. 252-253, 1970. (5) E. Hille, “Analytic Function Theory”, Vol. 1, Ginn & Co., Boston, Mass., 1959. (6) E. I. Jury, “Inners and Stability of Dynamic Systems”, Wiley, New York, 1974. (7) A. Leonhard, “Relative damping as a criterion for stability and an aid in finding the roots of a Hurwitz polynomial”, in “Automatic and Manual Control”, Butterworth, London, 19.52. (8) M. Marden, “Geometry of Polynomials”, American Mathematical Society, Providence, RI, 1966. (9) R. A. Silverman “Introductory Complex Analysis”, Dover, New York, 1972. (10)M. R. Stojic and D. D. Siljak, “Generalization of Hurwitz, Nyquist, and Mikhailov stability criteria”, IEEE Trans. Aurom. Control, AC-l& pp. 250-255, 1965.
496
Journal of The Franklin Institute Pergamon Press Ltd.
T. A. BICKART
of Electricul
Department Syracuse, and
and
Computer
Engineering,
Syracuse
University,
New York 13210
E. I. JURY
Department Electronics
of Electrical Engineering and Computer Research Laboratory, University of California,
Sciences Berkeley,
and
the
California
94720
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. 7’he number of tests needed is at most 1 + [(In k)/(ln 3)1, where k is the integer associated with the central angle r/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 application of a collection of mappings on the complex plane defined by a particular collection of Schwarz-Christoflel transformations. ABSTRACT:
I. Introduction 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 [(6) p. 23, (7), (lo)]. The degree of relative stability is (Y,where 2cx is the central angle of the
sector. 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 consideringrelative stability of degree a-can be placed in an abstract framework. This is accomplished using Definition : A polynomial P is said to be an R[a]-polynomial if for some CYE (0, m/2] ull rhe zeros of P are in the sector .Yw= {s: /arg {-.~}I <(Y}. Polynomial root clustering in a sector has been considered by others [(l), (3), (4), (6) pp. 23-27, (8) pp. 191-193, (lo)]. 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 Schwarz-Christoffel transformation (3). Specifically, we shall describe a test for R[a]-stability when a = r/2k
t Research sponsored in part by a special grant from the Syracuse CJniversity Senate Research Committee to the first author and by the National Science Foundation under Grant ENG76-21816 to the second author.
0 The Franklin Institute001~032/79/1101~87$02.00/0
487
7’. A. Bickart
and E. 1. Jury
Re s
FIG. 1. Sector in left half-plane.
(k = 1,2,...)
requiring N = 1+ [(ln k)/(ln 3)]
tests on a polynomial
to be strictly
(1)
Hurwitz.?
II. Principal Result The Schwarz-Christoffel
transformation
[(9) pp. 332-3391
s = -(-p)“” maps
each of the sectors %,?n
onto
(2)
the open
= {s: larg {-sei*W”m>j < ~/2m)
(I = 0, . . . , m - 1)
(3)
left half-plane Z’={(p:
Rep
(4)
These sectors for m = 1 , . . . ,4 are depicted in Fig. 2. Let PI denote the polynomial P,(s) = P(seiz?rum) (l= O,...,m - 1). Then, rather obviously, all the zeros of P = PO are in YW,2m= SPzjlZ,,,if, and only if, all the zeros of Pt are in P’&,,. Now set P” = rI;‘,-,‘P(
(5)
and x%n
= u i%1%2m.
(6)
Note : Y:Qm is the preimage by the transformation (2) of 2. Clearly all the zeros of P are in Ymlzm only if all the zeros of P” are in Y’J$,,. Unfortunately t The notation [rj. r a real number, is used to signify the least integer greater than or equal to r. Alternatively, [rJ is used to signify the greatest integer less than or equal to r.
488
Journal ofTheFranklin Institute Pergamon PressLtd.
Polynomial Root Clustering Im
5
Im s
Re s
Re s
m-2
m=l Im
s
Im s
Re s
Re s
m=/l
m=3
FIG. 2. Sectors Lf?&,,,(e = 0,. . . , m - 1; m = 1,2,3,4).
the converse stated in
relation is not true. However,
a reciprocal
relation does exist, as
Theorem 1. Let z& ={s: P”(s) = 0)
(7)
K = [(ln k)/(ln 3)J.
(8)
and Then P is an R[r/2k]-polynomial
if, and only if,
A m._MkEn = xbII> where A denotes the logical “and” operation A& ={l,
(9)
and
3,9 ,..., 3K}U{k}.
(10)
Proof: [only if] Let e={S:P(s)=O}. Suppose P is an 2 = yn/&n for all Y”_,*,,,. Therefore [if] Obviously
(11)
R[r/2k]-polynomial; that is, suppose 5%t Y’m,Zk.Then clearly m E .I%,. But, as previously observed, 3 c YmIZ,,,only if %, c (9) must be true. if Z, c .P’&,,, then ZEc .YFlzm. Hence, by (9), we know that 3 = l-l mE”uA(t~%n.
Vol. 308, No. 5, November 1979 Printed in Northern Ireland
(12) 489
T. A. Bickart and E. I. Jury Now, as can be easily shown,
n rn&uk xkrn=elk.
It therefore
follows that % t .Y,,,2k;equivalently,
(13)
P is an R[?r/2k]-polynomial.
0 Suppose K and k were not related as in (8). In that case it would be easy to show, for all non-negative integers K, that (13) would be valid when 3K 5 k I 3 K+* and would not be valid when 3Ki1 < k. It therefore follows with K defined as in (8) that, as a function of k, the rate of increase of the cardinality of JU, is as small as possible. We must now consider the problem of validating each statement of (9); that is, of validating %n = x&m (m ~JUThe needed criterion is to be found in Theorem II.
(14)
Theorem II. The statement %?, c XLn
(1%
Q”(p) = P”(s) Js-+Iyp
(16)
is true if, and only if,
is a strictly Hurwitz polynomial. Proof: Observe, from the manner by which Pk is defined in (5) that the transformation (2) applied to P” yields a polynomial in p: P”[-(-p)““‘]
= P’“(s)),~~=(_~)~+L~ = Q”(p).
(17)
Note: degree {P(s)} =degree {Q”‘(p)}. [only $1 As (2) maps Y&,, onto 2 and as 2 c LY~,~,,,it follows that the zeros of Q”(p) are in Z-Q”(p) is a strictly Hurwitz polynomial. [if] As the preimage of any point of 2 by (2) is in 0 x&m and as the zeros of Q”‘(p) are in 2, it follows that Z,,, c .Y$,,,. As an immediate consequence of Theorems I and II and the fact that N of (1) is the cardinality of the set .&, P can be shown to be or not to be an R[r/2k]-polynomial by carrying out N tests for a polynomial to be a strictly Hurwitz polynomial. This procedure is presented as the R[rr/2k]-stability test: P is an R[m/2k]-polynomial if, and only if, for all m E A,,
Q” is a strictly Hurwitz polynomial.
As an illustration of this result, consider the polynomial P(s)=s3+7sz+16s+10. To determine wheather it is an R[?r/8]-polynomial (k = 4) we must show that each of the following three polynomials is a strictly Hurwitz pOlynOmial: Q'(p)=s3+7~2+16s+10),=,=p3+7p2+16p+10, Q3(p)=s9+
37s"+1036s3+1000),+,=p3+37p2-+1036p+1000,
Q4(p)=-~1Z+57s8-10056s4+10000~,~=_,, =p3+.57p2+10056p+100000. 490
Polynomial Root Clustering
By the Lienard-Chipart criterion [(6) pp. 22-231, each of these polynomials is strictly Hurwitz. Therefore P(s) is an R[r/8]-polynomial. III. Secondary Results If the constituent
test corresponding
to m = 1:
Q’(p) is a strictly Hurwitz polynomial, is replaced by: Q ‘(p - a) is a strictly Hurwitz polynomial with 0 I a, then for an affirmative test, P(s) will have all its zeros not in .YPrrlZk but in .Ym,2k truncated on the right at s =-a, as depicted in Fig. 3(a). If the replacement is by: Ql(p -a)
and Q’(-p -b) are strictly Hurwitz polynomials with 05 a 5 b,
then for an affirmative test, the zeros of P(s) will be in 5p?r/2ktruncated on the right at s = -a and on the left at s = -b, as depicted in Fig. 3(b). Both of these results are easily verified. By applying the R[r/2k]-stability test to the polynomial P(s-d) for some real d, an affirmative test establishes the region of root clustering to be Ym,2k translated to the left by d, as depicted in Fig. 4(a). When the constituent test for m = 1 is also replaced as discussed above, with d 5 a I b, the region is that depicted in Fig. 4(b).
Re s
Re s
ia)
ib) FIG. 3. Truncated
sectors.
Re s
Re s
(a1
(b)
FIG. 4. Translated Vol. 308, No. 5, November 1979 Printedin Northern Ireland
sectors.
491
T. A. Bickart and E. I. Jury In the preceding paragraph we indicated how, by simple alteration of the R[m/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. Even more can be achieved as we next illustrate through several examples. IV. Examples If it can be shown that Q’(p -a) with Osa and a’(--p) are strictly Hurwitz polynomials, then the roots of P(s) are confined to a pair of sectors symmetric with respect to the real axis and truncated on the right, as depicted in Fig. 5(a). On the other hand, if, corresponding to the polynomial P(s -d), it can be shown that Q’(p - a) with d CCa, a’(--p - b) with a < b, and Q”(-p) are strictly Hurwitz polynomials, then the roots of P(s) 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. 5(b). As a last example, suppose it can be shown that Q’(p), Q”(p), and a”(-p) are strictly Hurwitz polynomials, then the roots of P(s) are confined to the pair of sectors depicted in Fig. 5(c). The variety of achievable regions is exceedingly great as the following general result makes clear: Let PC” (s) = P(qs + /3,); suppose Q(l*m)is the m-th = polynomial derived from PC’) as in (5). Then the zeros of P are in Sp(‘Vm7n) (s: (s - &)“’ + (- q)“’ (~,p + A,,) = 0 and p E .Z’e)if, and only if, Q(l,m.“)(p) = Q(*,“‘) ( K~P+ A,) is a strictly Hurwitz polynomial in p. This result is a simple extension of Theorem II-the new features are the linear transformations of the s-plane
Re s
Re s
I
(a) Im s
Re s
FIG. 5. Pairs of sectors. 492
(,b)
Polynomial Root CZustering
before and of the p-plane after invoking the Schwarz-Christoffel transformation. The counterpart of the R[v/2k]-stability test (Theorem II with Theorem I) is: The zeros of P are in n(l,m,njod Y (‘Jw) if, and only if, Q(‘rm*“)is a strictly Hun&z polynomial for all (l,m,n)Ed, d being a suitable set of triplets of index values. The number of constituent tests for a polynomial to be a strictly Hurwitz polynominal is the cardinality of ~4. To further illustrate this rather general conclusion, we cite two cases. Suppose Q(l,m*“)is a strictly Hurwitz polymonial for each (l,m,n)E.d, where with b 50
(6 m, n) aI 6
d
K,
A,
(l,l,l)
1
0
1 0
(1,3,1>
1
0
1 0
(2,1,2)
1 -b
-1
0
(2,3,2)
1 -b
-1
0.
Then the roots of P are in the diamond shaped region depicted in Fig. 6(a). As a small variation on this case, suppose Q(l.“‘+) is a strictly Hurwitz polynominal for each (l,m,n) E Se, where with b > a > 0 (1, m, n) (~1 PI (1,L
se
4
A,
1) 1
0
1
0
(1,3,1)
1
0
1
0
(2,1,2)
1 -b
(1,3,3)
1
-1
0
0 -1 -a3.
Then the roots of P are in the not-diamond shaped region depicted in Fig. 6(b). There is a further level of generalization which subsumes the former. Therefore, we will adopt similar notation in briefly describing it. Let q denote the degree of P. Set P(I) (s) = (ns + 6,)” P[(qs + f$)/(y,s + SJ], where q and 7/l are such that q/n is not a zero of P. (It then follows that the degree of PC’) is also 77.)Suppose Q(‘,“‘) is the m-th polynomial derived from P(l) as in (5). Set
Im s
Ims
Res
Re s
(a)
(b)
FIG.6. Regions of root clustering. Vol. 308, No. 5, Novemkr 1979 Printed in Northern Ireland
493
Q(h,n)
(p) = (p,p + u,)” Q’**“’[(~,p + h,)/(&p + v,,)], where K,, and p,,, are such that K,,/&, is not a zero of Q ‘Lm:. Now the zeros of P are in Lt’((“‘,“)= {s:(&s-fir)‘” (p+,p+~,,)+(yrs-a[)” (~,p+h,)=O and p~2’} if, and only if, Q(P,m,n)is a strictly Hurwitz polynomial. This simple extension of Theorem II incorporates bilinear (or Mobius) transformations [(S) pp. 46-581 of the s-plane before and of the p-plane after invoking the Schwarz-Christoffel transformation. (The degree preserving conditions on q and -yl and on K, and CL, guarantee that each (finite) zero is transformed into a (finite) zero.) As above, the R[rr/2k]-stability test counterpart is: The zeros of P are in if, and only if, Q(‘.m*n)1s . a strictly Hurwitz polynomial for all (I,m,n) E d. To illustrate this very general result, we offer the following two examples: Suppose Q”,“‘+) is a strictly Hurwitz polynomial for each (I,m,n) E d, where with a > 0
C/f, m*“) n(Lm.nk~
d
(1, Wr) o[ (l,l, 1) -a
Pr ?‘I 8, ‘G A, P, % 0 1 -1 1 0 0 1
(1,3,1)-a
0
1 -1
1 0
0
1.
Then the roots of P are in the tear-drop shaped region depicted in Fig. 7(a). Next suppose Q(r+.“) is a strictly Hurwitz polynomial for each (f,m,n) E Se, where with a > 0
(l,l,l) sQ
(2,3,1)-a
10
0 0
11
1 -1
0
0
1
1 0
0
1.
Then the roots of P are in the not-simply-connected 7(b).
region depicted in Fig.
(b)
FIG. 7. Regions of root clustering.
494
Polynomial Root Clustering
V. 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 (Ye,PI, -yl,and 8, and of K,, A,, CL,, and v,, 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 YF,,-,,,- 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 n,,,,,,,,, Z@“‘+). We began this paper by focussing on sectors because such regions are easily categorized. This quite 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(I,m,n)Ed .9”f3m+).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 cause, we solved that problem by creating the polynomial am(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 Q” was just that of P. In another paper we examine the root clustering problem by composing the complement of the region as a union of disks and half-planes (2). 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 (1) B. D. 0. Anderson, N. K. Bose and E. I. Jury, “A simple test for zeros of a complex polynomial in a sector”, IEEE Trans. Autom. Control. AC-19, pp. 437-438, 1974. (2) T. A. Bickart and E. I. Jury, “Regions of polynomial root clustering”, J. Franklin Inst., Vol. 304, pp. 149-160, 1977. Vol. 308, No. 5. November 1979 F’rintedin Northern Ireland
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T. A. Bickart
and E. I. Jury
(3)N. K. Bose and E. I. Jury, “The
Schwarz-Christoffel transformation applied to stability problems”, Proc. Asilomar Conf. on Circuits and Systems, pp. 148-152, 1973. (4) E. J. Davison and N. Ramesh, “A note on the eigenvalues of a real matrix”, IEEE Trans. Autom. Control., AC-15, pp. 252-253, 1970. (5) E. Hille, “Analytic Function Theory”, Vol. 1, Ginn & Co., Boston, Mass., 1959. (6) E. I. Jury, “Inners and Stability of Dynamic Systems”, Wiley, New York, 1974. (7) A. Leonhard, “Relative damping as a criterion for stability and an aid in finding the roots of a Hurwitz polynomial”, in “Automatic and Manual Control”, Butterworth, London, 19.52. (8) M. Marden, “Geometry of Polynomials”, American Mathematical Society, Providence, RI, 1966. (9) R. A. Silverman “Introductory Complex Analysis”, Dover, New York, 1972. (10)M. R. Stojic and D. D. Siljak, “Generalization of Hurwitz, Nyquist, and Mikhailov stability criteria”, IEEE Trans. Aurom. Control, AC-l& pp. 250-255, 1965.
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Journal of The Franklin Institute Pergamon Press Ltd.