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On the Conditions for the Characteristic Roots of a Matrix in a Sector
On the Conditions for the Characteristic Roots of a A4atrix in a Sector by TAKEHIRO
MORI
Automation Research Laboratory, Gokasho, Uji, Kyoto 611, Japan
Faculty
of Engineering,
Kyoto
University,
Kyoto
University,
NOR10 FUKUMA
Kyoto University of Education, Fushimi-ku, Kyoto 612, Japan and
MICHIYOSHI
KUWAHARA
Automation Research Laboratory, Gokasho, Uji, Kyoto 611, Japan
ABSTRACT :
Davison and Ramesh
Faculty
expressed
of Engineering,
equivalently
the condition for the characteristic
roots of a real matrix to lie within a sector by the condition for a matrix of doubled order to be Hurwitzian
(1). A further generalization
of their result is given in this note.
Z. Introduction Davison and Ramesh (1) proved that a necessary and sufficient condition for the characteristic roots of a matrix A E R” xn to lie inside the hatched region of Fig. 1 is that the matrix of order 2n given by A;(& PI L
(A + pl,) cos 6 [ (A+pZ,)sin
6
-(A + pl,) sin 6 (A+pZ,)cos
6
(1)
1’
where I, is the identity matrix of order n, is a Hurwitzian, i.e. the real parts of the characteristic roots are all negative (1). We hereafter denote the class of Hurwitzian by X. In this note, some generalized results concerning this subject are given.
ZZ.Main Results Theorem 1 The characteristic roots of A E R” xn, n,(A) lie inside the sector PQP in Fig. 1 (left hand side of lines PQ and Qp), iff A;(4 0) + (p cos 6 -g
sin 6)Z,, E #
(2)
n
Proof: Davison and Ramesh’s result is based on the fact that the characteristic roots of A;(6, p) are 1i(A + pl,) e *j’. (Though this was not explicitly proven in (l), this 0
The Franklin Institute 00I~32/83/02008743%03.00/0
87
T. Mori, N. Fukuma
and M. Kuwahara
FIG. 1. Specified sectors in the complex plane.
is easily confirmed by noting that A’i(6,p) is similarly ((A +pl,) e+j’, (A +pZ,)e-j6) by the unitary matrix,
transformed
to block diag
Other proofs are given in (2), (4). Now, by the fact that in general the point in the complex plane obtained by rotating a point z by angle f 6 around some other point a can be expressed by a+(z--a)e’j’ = z e ‘js + (1 -e’ j’)u, the requirement of the theorem can be written as Re [&(A)e*@+(l
-e*j”)a]+p
< 0,
a = -p+jq
(3)
or Re [n,(A) e ‘j’] + (p cos 6 - q sin 6) < 0. Referring
to Davison
and Ramesh’s result, (4) is equivalently
(4) rewritten
as (2).
QED Now, since the co-ordinate immediately yields : Corollary
of the point Q in Fig. 1 is (q tan 6 - p, 0), Theorem
I
1 A;(&p-q
tan ~)EP
(5)
iff A”,(& 0) + (p cos ??I,, - q sin 6Z,,) E 2”.
(6)
Specifically, putting q = 0 in (5) leads to the condition by Davison by Corollary 1 we have another expression of their result :
H
and Ramesh, and
Corollary 2
All the characteristic
roots of A exist inside the hatched A?(& 0) + p cos 6Z,, E 3.
88
region in Fig. 1 iff (7)
n
.lourna, of the Frankhn lnat~tute Press Ltd.
F’ergamon
On the Conditionsjbr
the Characteristic
Roots of a Matrix
in a Sector
The condition (7) would be more tractable than A;(6, p) E Z’, since p is included only in the diagonal block matrices. Finally, we give a condition under which Ai(A exist in the non-symmetric region with respect to the real axis in the complex plane. Such a problem occurs, say, when the describing function approach is employed for the analysis and synthesis of systems with memory-type nonlinearity. Corollary 3 A necessary and sufficient condition for &(A) of a complex the sector having the point P( --p, q) as the vertex and making imaginary axis (Fig, 1) is
x:“(d)
E
(8)
A?“,
where X:“(a) is the real square matrix of order 4n constructed
x34 =
sin
by
-X”sin6 X”cos61
X” cos 6 x”
matrix A to be inside an angle of 6 with the
($
[
(9)
and X”
zz
Im (4 - d,
Re(4 + ~1,
Re(A)+pl,
-Im(A)+ql, Proqf: In view of the proof of Theorem expressed as Re [(&(A)-a)e’j’]
1, the statement
< 0,
(10) n
1’
a = -p+jq.
of Corollary
3 can be
(11)
This condition is the one for the complex matrix A-al, to have its characteristic roots in the sector having the vertex on the origin and angle of 6 with the imaginary axis. This has already been obtained in (3) and is equivalently written as (8).
QED III. Conclusions The conditions for the characteristic roots of a matrix to lie within the prescribed sector are considered. Davison and Ramesh’s result is recast and somewhat more generalized results are obtained. References (I) E. J. Davison and N. Ramesh, “A note on the eigenvalues of a real matrix”, IEEE Trans. automat. Contr., Vol. AC-15, pp. 252-253, 1970. (2) 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. automat. Contr., Vol. AC-19, pp. 437438, 1974. (3) B. D. 0. Anderson, N. K. Bose and E. I. Jury, “On eigenvalues of complex matrices in a sector”, IEEE Trans. automat. Contr., Vol. AC-20, p. 433, 1975. (4) J. W. Brewer and E. Aghavarari, “On the design of feedback control for relative stability”, Automatica, Vol. 17, pp. 771-772, 1981.
89
MORI
Automation Research Laboratory, Gokasho, Uji, Kyoto 611, Japan
Faculty
of Engineering,
Kyoto
University,
Kyoto
University,
NOR10 FUKUMA
Kyoto University of Education, Fushimi-ku, Kyoto 612, Japan and
MICHIYOSHI
KUWAHARA
Automation Research Laboratory, Gokasho, Uji, Kyoto 611, Japan
ABSTRACT :
Davison and Ramesh
Faculty
expressed
of Engineering,
equivalently
the condition for the characteristic
roots of a real matrix to lie within a sector by the condition for a matrix of doubled order to be Hurwitzian
(1). A further generalization
of their result is given in this note.
Z. Introduction Davison and Ramesh (1) proved that a necessary and sufficient condition for the characteristic roots of a matrix A E R” xn to lie inside the hatched region of Fig. 1 is that the matrix of order 2n given by A;(& PI L
(A + pl,) cos 6 [ (A+pZ,)sin
6
-(A + pl,) sin 6 (A+pZ,)cos
6
(1)
1’
where I, is the identity matrix of order n, is a Hurwitzian, i.e. the real parts of the characteristic roots are all negative (1). We hereafter denote the class of Hurwitzian by X. In this note, some generalized results concerning this subject are given.
ZZ.Main Results Theorem 1 The characteristic roots of A E R” xn, n,(A) lie inside the sector PQP in Fig. 1 (left hand side of lines PQ and Qp), iff A;(4 0) + (p cos 6 -g
sin 6)Z,, E #
(2)
n
Proof: Davison and Ramesh’s result is based on the fact that the characteristic roots of A;(6, p) are 1i(A + pl,) e *j’. (Though this was not explicitly proven in (l), this 0
The Franklin Institute 00I~32/83/02008743%03.00/0
87
T. Mori, N. Fukuma
and M. Kuwahara
FIG. 1. Specified sectors in the complex plane.
is easily confirmed by noting that A’i(6,p) is similarly ((A +pl,) e+j’, (A +pZ,)e-j6) by the unitary matrix,
transformed
to block diag
Other proofs are given in (2), (4). Now, by the fact that in general the point in the complex plane obtained by rotating a point z by angle f 6 around some other point a can be expressed by a+(z--a)e’j’ = z e ‘js + (1 -e’ j’)u, the requirement of the theorem can be written as Re [&(A)e*@+(l
-e*j”)a]+p
< 0,
a = -p+jq
(3)
or Re [n,(A) e ‘j’] + (p cos 6 - q sin 6) < 0. Referring
to Davison
and Ramesh’s result, (4) is equivalently
(4) rewritten
as (2).
QED Now, since the co-ordinate immediately yields : Corollary
of the point Q in Fig. 1 is (q tan 6 - p, 0), Theorem
I
1 A;(&p-q
tan ~)EP
(5)
iff A”,(& 0) + (p cos ??I,, - q sin 6Z,,) E 2”.
(6)
Specifically, putting q = 0 in (5) leads to the condition by Davison by Corollary 1 we have another expression of their result :
H
and Ramesh, and
Corollary 2
All the characteristic
roots of A exist inside the hatched A?(& 0) + p cos 6Z,, E 3.
88
region in Fig. 1 iff (7)
n
.lourna, of the Frankhn lnat~tute Press Ltd.
F’ergamon
On the Conditionsjbr
the Characteristic
Roots of a Matrix
in a Sector
The condition (7) would be more tractable than A;(6, p) E Z’, since p is included only in the diagonal block matrices. Finally, we give a condition under which Ai(A exist in the non-symmetric region with respect to the real axis in the complex plane. Such a problem occurs, say, when the describing function approach is employed for the analysis and synthesis of systems with memory-type nonlinearity. Corollary 3 A necessary and sufficient condition for &(A) of a complex the sector having the point P( --p, q) as the vertex and making imaginary axis (Fig, 1) is
x:“(d)
E
(8)
A?“,
where X:“(a) is the real square matrix of order 4n constructed
x34 =
sin
by
-X”sin6 X”cos61
X” cos 6 x”
matrix A to be inside an angle of 6 with the
($
[
(9)
and X”
zz
Im (4 - d,
Re(4 + ~1,
Re(A)+pl,
-Im(A)+ql, Proqf: In view of the proof of Theorem expressed as Re [(&(A)-a)e’j’]
1, the statement
< 0,
(10) n
1’
a = -p+jq.
of Corollary
3 can be
(11)
This condition is the one for the complex matrix A-al, to have its characteristic roots in the sector having the vertex on the origin and angle of 6 with the imaginary axis. This has already been obtained in (3) and is equivalently written as (8).
QED III. Conclusions The conditions for the characteristic roots of a matrix to lie within the prescribed sector are considered. Davison and Ramesh’s result is recast and somewhat more generalized results are obtained. References (I) E. J. Davison and N. Ramesh, “A note on the eigenvalues of a real matrix”, IEEE Trans. automat. Contr., Vol. AC-15, pp. 252-253, 1970. (2) 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. automat. Contr., Vol. AC-19, pp. 437438, 1974. (3) B. D. 0. Anderson, N. K. Bose and E. I. Jury, “On eigenvalues of complex matrices in a sector”, IEEE Trans. automat. Contr., Vol. AC-20, p. 433, 1975. (4) J. W. Brewer and E. Aghavarari, “On the design of feedback control for relative stability”, Automatica, Vol. 17, pp. 771-772, 1981.
89