Systems & Control Letters 57 (2008) 1058–1066
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Systems & Control Letters
journal homepage: www.elsevier.com/locate/sysconle
Numerical J -spectral factorization of general para-hermitian matrices
Jovan Stefanovski
∗
Control & Informatics Div., JP ‘‘Streževo’’, Bitola, Former Yugolav Republic of Macedonia, The
article info
Article history:
Received 17 December 2007
Received in revised form
9 April 2008
Accepted 13 July 2008
Available online 15 August 2008
Keywords:
Para-hermitian matrices
J -spectral factorization
Zeros at infinity
abstract
A numerical algorithm for J -spectral factorization of para-hermitian rational matrices is presented, based
on a generalization of the notion of separated realization of a para-hermitian matrix. The transformations
used preserve the para-hermitian matrix pencil structure.
© 2008 Elsevier B.V. All rights reserved.
1. Introduction
Consider a possibly singular, with possible zeros on the
imaginary axis (including the infinity), and possibly indefinite on
the imaginary axis m × m-dimensional real rational matrix Π (λ),
which is para-hermitian, i.e. Π
T
(−λ) = Π (λ). A problem of J -
spectral factorization is to find a real square rational matrix Φ(λ)
nonsingular for all R[λ] > 0,λ ∈ C, and indices m
+
, m
0
and m
−
satisfying m
+
+ m
0
+ m
−
= m, such that
Π (λ) = Φ
T
(−λ)J Φ(λ), J = diag{I
m
+
, 0
m
0
×m
0
, −I
m
−
}. (1.1)
The J -spectral factorization formulated in its most general form has
many applications (see [7,5] and references therein).
Another variants of the problem is to find a possibly nonsquare
Φ (1.1) and J = diag{I
m
+
, −I
m
−
}, such that Φ has full row rank for
all R[λ] > 0, and to find spectral factor Φ such that, besides of full
rank in R[λ] > 0, it is analytic in that region.
Notation and some standard results. By the superscript T we
denote matrix transposition. By the superscript ∗ we denote
conjugate transpose. The identity matrix is denoted by I , or I
n
if the
matrix dimension requires. The sets of complex and real numbers
are denoted by C and R. By null(K ) we denote a right annihilator of
matrix K .
The abbreviation FGE means finite generalized eigenvalue, and
the abbreviation IGE means infinite generalized eigenvalue.
∗
Tel.: +389 47207838; fax: +389 47207836.
E-mail address: jovanstef@t-home.mk.
Kronecker canonical form of matrix pencil λM − N is:
P (λM − N )Q = diag
[λI
n
r
− A
c
, −B
c
],λM
∞
− I ,
λI
n
f
− J
f
,
λI
n
l
− A
o
−C
o
(1.2)
where (A
c
, B
c
) is a controllable pair, (A
o
, C
o
) is observable, M
∞
is
a nilpotent matrix of the form M
∞
= diag{M
∞i
, i = 1,..., s},
where M
∞i
=
0 I
ν
i
−1
0 0
are ν
i
× ν
i
-dimensional matrices. The
indices ν
i
, i = 1,..., s are multiplicities of the IGEs. J
f
is a matrix
that contains the FGEs of λM − N and P and Q are non-singular
matrices [20], or Kronecker canonical form is
P (λM − N )Q =
0 0
0 λM
r
− N
r
, (1.3)
where λM
r
− N
r
is a regular matrix pencil.
Matrix pencil λM − N is para-hermitian if M
T
= −M and
N
T
= N . If λM − N is para-hermitian then n
l
= n
r
in (1.2), and the
FGEs of λM − N that are not purely imaginary appear in symmetric
pairs, in respect to imaginary axis.
The zeros of a rational matrix are defined through its McMillan
form. A rational matrix is minimum phase if it has no zeros in
R[λ] > 0. A rational matrix is stable if it is analytic in R[λ] > 0.
A square matrix is stable if it has no eigenvalues in R[λ] > 0.
Analogously we define stable FGEs.
The following notions and results are taken from [13]. The
descriptor realization (E , A, B, C , D) of transfer matrix G(λ) =
D + C (λE − A)
−1
B is weakly minimal if and only if matrices
[λE − A, −B] and [E , B] have full row ranks (finite and infinite
0167-6911/$ – see front matter © 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.sysconle.2008.07.002