1. Introduction
Consider a linear time invariant system
uid1
to be identified with the pair of matrices where ,
and or the fields of the real or complex
numbers. If state-feedback is applied to system (▭),
Rosenbrock's Theorem on pole assignment (see [1]) characterizes for
the closed-loop system
uid2
the invariant factors of its state-space matrix .
This result can be seen as the solution of an inverse problem; that of finding
a non-singular polynomial matrix with prescribed invariant factors and
left Wiener–Hopf factorization indices at infinity. To see this we recall that
the invariant factors form a complete system of
invariants for the finite equivalence of polynomial matrices (this
equivalence relation will be revisited in Section ▭) and it will be seen in Section
▭ that any polynomial matrix is left Wiener–Hopf equivalent at infinity to a
diagonal matrix , where the non-negative integers
(that can be assumed in non-increasing order) form a
complete system of invariants for the left Wiener–Hopf equivalence
at infinity. Consider now the transfer function matrix of
(▭). This is a rational matrix that can be written as an irreducible
matrix fraction description , where and
are right coprime polynomial matrices. In the terminology of
[2], is a polynomial matrix representation of
(▭), concept that is closely related to that of polynomial
model introduced by Fuhrmann (see for example [3] and the references
therein). It turns out that all polynomial matrix representations of a system are right
equivalent (see [2], [3]), that is, if and are polynomial
matrix representations of the same system there exists a unimodular matrix
such that . Therefore
all polynomial matrix representations of (▭) have the same invariant
factors, which are the invariant factors of except for some trivial ones.
Furthermore, all polynomial matrix representations also have the same left Wiener–
Hopf factorization indices at infinity, which are equal to the controllability indices of
(▭) and (▭), because the controllability indices are
invariant under feedback. With all this in mind it is not hard to see that Rosenbrock's
Theorem on pole assignment is equivalent to finding necessary and sufficient conditions
for the existence of a non-singular polynomial matrix with prescribed invariant
factors and left Wiener–Hopf factorization indices at infinity. This result will be
precisely stated in Section ▭ once all the elements that appear are properly
defined. In addition, there is a similar result to Rosenbrock's Theorem
on pole assignment but involving the infinite structure (see [4]).
Our goal is to generalize both results (the finite and infinite versions of Rosenbrock's
Theorem) for rational matrices defined on arbitrary fields via local rings. This will be
done in Section ▭ and an extension to arbitrary fields of the concept of Wiener–Hopf equivalence will be needed. This concept is very well established for complex
valued rational matrix functions (see for example [5], [6]). Originally
it requires a closed contour, , that divides the extended
complex plane () into two parts: the inner domain ()
and the region outside (), which contains the point at infinity.
Then two non-singular complex rational matrices and ,
with no poles and no zeros in , are said to be left Wiener–Hopf equivalent with
respect to if there are matrices and
with no poles and no zeros in and ,
respectively, such that
uid3
It can be seen, then, that any non-singular complex rational matrix
is left Wiener–Hopf equivalent with respect to to a diagonal matrix
uid4
where is any complex number in and
are integers uniquely determined by . They are called the left Wiener–Hopf
factorization indices of with respect to (see again [5], [6]). The generalization to arbitrary fields relies on the following idea:
We can identify and
with two sets and ,
respectively, of maximal ideals of . In fact, to each
we associate the ideal generated by , which is a
maximal ideal of . Notice that is also a prime
polynomial of but and , as defined, cannot contain
the zero ideal, which is prime. Thus we are led to consider the set
of maximal ideals of . By using this
identification we define the left Wiener–Hopf equivalence of
rational matrices over an arbitrary field with respect to a
subset of , the set of all maximal ideals of
. In this study local rings play a fundamental role. They
will be introduced in Section ▭. Localization techniques have been used previously in the algebraic theory of linear systems (see, for example, [7]). In Section ▭ the algebraic structure
of the rings of proper rational functions with prescribed finite poles is studied (i.e., for a
fixed the ring of proper rational functions
with for
all ). It will be shown that if there is an ideal
generated by a linear polynomial outside then the set of proper rational
functions with no poles in is an Euclidean domain and all rational matrices can
be classified according to their Smith–McMillan invariants.
In this case, two types of invariants live together for any non-singular rational matrix
and any set : its Smith–McMillan and left Wiener–Hopf
invariants. In Section ▭ we show that a Rosenbrock-like Theorem
holds true that completely characterizes the relationship between these two types of
invariants.
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2. Preliminaries
In the sequel will denote the ring of polynomials with coefficients in an arbitrary field and the set of all maximal ideals of , that is,
uid5
Let be a monic irreducible non-constant polynomial. Let be the multiplicative subset of
whose elements are coprime with . We denote by
the quotient ring of by ; i.e.,
:
uid6
This is the localization of at (see [8]).
The units of are the rational functions
such that and
. Consequentially,
uid7
For any , let
uid8
This is a ring whose units are the rational functions
such that for all ideals ,
and .
Notice that, in particular, if then and if then
, the field of rational functions.
Moreover, if is a non-constant polynomial
whose prime factorization, , satisfies the condition that for all , we will say that factorizes in or
has all its zeros in . We will consider that the only
polynomials that factorize in are the constants. We
say that a non-zero rational function factorizes in if both its
numerator and denominator factorize in . In this case we will say
that the rational function has all its zeros and poles in .
Similarly, we will say that has no poles in
if and for all ideals
. And it has no zeros in if
for all ideals . In other words, it is equivalent
that has no poles and no zeros in and that
is a unit of . So, a non-zero
rational function factorizes in if and only if it is a unit in
.
Let denote the set of matrices with elements in . A matrix is invertible in if all its elements are in and its determinant is a unit in . We denote by the group of units of .
Remark 1
Let . Notice that
1. If then and .
2. and .
For any the ring is a principal ideal domain (see [9]) and its field of fractions is . Two matrices are equivalent with respect to if there exist matrices such that .
Since is a principal ideal domain, for all non-singular (see [10]) there exist matrices
such that
uid10
with (“” stands for
divisibility) monic polynomials factorizing in , unique up to multiplication by units of . The diagonal matrix is the Smith normal form of with respect to and are called the invariant factors of with respect to . Now we introduce the Smith–McMillan form with respect to . Assume that is a non-singular rational matrix. Then
with and monic, factorizing
in . Let be
the Smith normal form with respect to of , i.e.,
invertible in and
monic polynomials factorizing in . Then
uid11
where are irreducible rational
functions, which are the result of dividing by and canceling the common
factors. They satisfy that ,
are monic polynomials factorizing in . The diagonal
matrix in (▭)
is the Smith–McMillan form with respect to .
The rational functions
, , are called the
invariant rational functions of with respect to and constitute a complete system of invariants of the equivalence with respect to for rational matrices.
In particular, if then
, the
matrices are unimodular matrices,
(▭) is the global Smith–McMillan form of a
rational matrix (see [11] or [1] when
or ) and are the global
invariant rational functions of .
From now on rational matrices will be assumed to be non-singular
unless the opposite is specified. Given any
we say that an non-singular
rational matrix has no zeros and no poles in if its global
invariant rational functions are units of . If its global
invariant rational functions factorize in , the matrix has its
global finite structure localized in and we say that the matrix
has all zeros and poles in . The former means that and the latter that because and are non-zero
constants. The following result clarifies the relationship between
the global finite structure of any rational matrix and its local
structure with respect to any .
Proposition 2 Let . Let be non-singular with
its global invariant rational functions
and let
be irreducible rational functions such that ,
are monic polynomials factorizing in . The following properties are equivalent:
There exist such that the global invariant
rational functions of are
,
and .
There exist matrices invertible in such that
uid15
i.e., are the invariant rational functions of with respect to .
and with units of , for .
Proof.- 1 2. Since the global invariant rational functions of are
, there exist such that
As , by Remark ▭.1, .
Therefore, putting and it follows that
and are invertible in and
.
2 3. There exist unimodular matrices
such that
uid17
with irreducible rational functions such that and are monic polynomials. Write
such that
factorize in and factorize in .
Then
uid18
with and
invertible in . Since the
Smith–McMillan form with respect to is unique we get that
.
3 1. Write (▭) as
uid19
It follows that with
and .
Corollary 3
Let be non-singular and such that . If are the invariant rational functions of
with respect to , , then
are the invariant rational functions of
with respect to .
1.08
Proof.- Let be the global invariant rational functions of .
By Proposition ▭,
with units of . On the other hand
with units of .
So, or equivalently
The polynomials are coprime because factorizes in , factorizes in and . In consequence and . Therefore, there exist polynomials , unit of , and , unit of , such that
Since and
. This implies that unit of . Following the same ideas we can prove that with a unit of . By Proposition ▭ are the invariant rational functions of
with respect to .
Corollary 4
Let . Two non-singular matrices are equivalent with respect to if and only if they are equivalent with respect to and with respect to .
Proof.- Notice that by Remark ▭.2 two matrices are equivalent with respect to if and only if there exist invertible in such that . Since and are invertible in both and then and are equivalent with respect to and with respect to .
Conversely, if and are equivalent with respect to and with respect to then, by the necessity of this result, they are equivalent with respect to
, with respect to and with respect to . Let be the invariant rational functions of and with respect to
, be the invariant rational functions of and with respect to
and be the invariant rational functions of and with respect to . By Corollary ▭ must be the invariant rational functions of and with respect to . Therefore, and are equivalent with respect to .
Let
be the ring of proper rational functions, that is,
rational functions with the degree of the numerator at most the
degree of the denominator. The units in this ring are the rational
functions whose numerators and denominators have the same degree.
They are called biproper rational functions. A matrix
is said to be biproper if
it is a unit in or, what is the same,
if its determinant is a biproper rational function.
Recall that a rational
function has a pole (zero) at if
has a pole (zero) at 0. Following
this idea, we can define the local ring at as the set of
rational functions, , such that
does not have 0 as a pole, that is,
. If
with , , where stands for “degree of”, then
uid22
As , then
uid23
Thus, this set is the ring of proper rational functions,
.
Two rational matrices are equivalent at infinity if there exist biproper matrices such that . Given a non-singular rational matrix (see [11]) there always exist
such that
uid24
where are integers. They are called the
invariant orders of at infinity and the rational functions are called the invariant rational functions of at infinity.
1.05
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3. Structure of the ring of proper rational functions
with prescribed finite poles
Let . Any non-zero rational function can be uniquely written as where is an irreducible rational function
factorizing in and is a unit of .
Define the following function over
(see [11], [12]):
uid25
This mapping is not a discrete valuation of if : Given two non-zero elements it is clear that
; but it may not satisfy that .
For example, let . Put
and .
We have that , but .
However, if and where , , the map
uid26
defined via if and if is a discrete valuation of .
Consider the subset of , , consisting of all proper rational functions with poles in , that is,
the elements of are proper
rational functions whose denominators are coprime with all the
polynomials such that . Notice that
if and only if where:
is a polynomial factorizing in ,
is an irreducible rational function and a unit of ,
or equivalently .
In particular implies that . The units in
are biproper rational functions
, that is , with factorizing in . Furthermore, is an integral domain whose field of fractions is provided that (see, for example,
Prop.5.22[11]). Notice that for , .
Assume that there are ideals in
generated by linear polynomials and let be any of them. The
elements of can be written as
where factorizes
in , is a unit in and
.
If is algebraically closed, for example , and
the previous condition is always fulfilled.
The divisibility in is characterized in the following lemma.
Lemma 5
Let . Let be such that
and
with factorizing in and irreducible rational functions, units of
. Then divides in
if and only if
uid31
uid32
Proof.- If then there exists
, with
factorizing in and
coprime, factorizing in , such that
. Equivalently,
.
So and
.
Moreover, as is a proper rational function,
and
.
Conversely, if then there is ,
factorizing in , such that . Write where
is an irreducible fraction representation of , i.e.,
after canceling possible common factors. Thus and
uid33
Then and .
Notice that condition (▭) means that in and condition (▭) means that in . So, in if and only if simultaneously in and .
Lemma 6
Let . Let be such that
and as in Lemma ▭. If and are
coprime in and either or then
and are coprime in .
Proof.- Suppose that and are not coprime.
Then there exists a non-unit such that and . As is not a unit, is not a constant or
. If is not a constant then and
which is impossible because and
are coprime. Otherwise, if is a constant then
and we have that and
. But this is again
impossible.
It follows from this Lemma that if are coprime in both rings and then are coprime in . The following example shows that the converse is not true in general.
Example 7
Suppose that and
. It is not difficult to
prove that and
are coprime elements in
.
Assume that there exists a non-unit
such that
and . Then
, and
. Since is not a unit, cannot be a constant. Hence,
, , and , but this is impossible because and are powers of . Therefore and must be coprime. However and
are not coprime.
Now, we have the following property when there are ideals in , , generated by linear polynomials.
Lemma 8
Let . Assume that there are ideals in
generated by linear polynomials and let be any of them. Let
be such that
and
. If and are
coprime in then and are coprime in and either
or .
Proof.- Suppose that and are not coprime in .
Then there exists a non-constant
such that and . Let .
Then is not a unit in
and divides and
because and . This is impossible, so and must be
coprime.
Now suppose that and . Let
. We have that . Thus
is not a unit in
and divides and
because and . This is
again impossible and either or
.
The above lemmas yield a characterization of coprimeness of elements in
when
excludes at least one ideal generated by a linear polynomial.
Following the same steps as in p. 11[12] and p. 271[11] we get the following result.
Lemma 9 Let and assume that there is at least an ideal in
generated by a linear polynomial. Then
is a Euclidean domain.
The following examples show that if all ideals generated by
polynomials of degree one are in , the ring
may not be a Bezout domain. Thus, it may not be
a Euclidean domain. Even more, it may not be a greatest common divisor
domain.
Example 10
Let and . Let
. We have seen, in the previous
example, that are coprime. We show now that the Bezout identity is not
fulfilled, that is, there are not such that
, with a unit in
. Elements in
are of the form
with relatively prime with
and and the units in are non-zero constants. We will
see that there are not elements ,
with and coprime
with , and such that
, with non-zero constant. Assume that
We conclude that or is a
multiple of , which is impossible.
Example 11
Let and . A fraction if and only if .
Let
.
By Lemma ▭:
.
If and then or with a non-zero constant.
Then and if and only if and or and .
So, the list of common divisors of and is:
uid42
If there would be a greatest common divisor, say , then because must be
a multiple of and . Thus such a greatest common divisor should be either or , but
does not divide neither of them because
uid43
Thus, and do not have greatest common divisor.
3.1. Smith–McMillan form
A matrix is invertible in
if and its determinant is a
unit in both rings, and , i.e., if and only if
.
Two matrices are
equivalent in if there exist
invertible in such that
uid45
If there are ideals in
generated by linear polynomials then is an Euclidean ring
and any matrix with elements in admits a
Smith normal form (see [10], [11] or [12]). Bearing in mind the
characterization of divisibility in given in Lemma
▭ we have
Theorem 12 (Smith normal form in )
Let . Assume that there are ideals in
generated by linear polynomials and let be one of them. Let be non-singular. Then there exist invertible in
such that
uid47
with monic polynomials factorizing in
and integers such that .
Under the hypothesis of the last theorem
form a complete system of invariants for the equivalence in
and are called the invariant rational
functions of in . Notice that
because divides .
Recall that the field of fractions of is
when . Thus we can talk about equivalence of matrix rational
functions.
Two rational matrices are
equivalent in if there are
invertible in
such that
uid48
When all ideals generated by linear polynomials are not in , each rational
matrix admits a reduction to Smith–McMillan form with respect to
.
Theorem 13
(Smith–McMillan form in
)
Let . Assume that there are ideals in
generated by linear polynomials and let be any of them. Let
be a non-singular matrix. Then there
exist invertible in such that
uid50
with coprime for all such that , are monic polynomials
factorizing in ,
divides for while
divides for .
The elements of the diagonal matrix, satisfying the conditions of the previous theorem,
constitute a complete system of invariant for the equivalence in of rational matrices. However, this system of invariants is
not minimal. A smaller one can be obtained by substituting each pair of positive
integers by its difference .
Theorem 14
Under the conditions of Theorem ▭, with , monic and coprime polynomials factorizing in ,
while and integers
such that
also constitute a complete system of invariants for the equivalence in .
Proof.- We only have to show that from the system , ,
satisfying the conditions of Theorem ▭, the system
, ,
can be constructed satisfying the conditions of Theorem ▭.
Suppose that , are monic and coprime polynomials factorizing
in such that and .
And suppose also that are integers such that
.
If for all , we define non-negative integers and for . If
for all , we define and .
Otherwise
there is an index such that
uid52
Define now the non-negative integers as follows:
uid53
Notice that . Moreover,
uid54
uid55
and using (▭), (▭)
uid56
uid57
In any case and
are elements of .
Now, on the one hand are coprime and
or . This means (Lemma ▭) that
are coprime for all
. On the other hand and
.
Then (Lemma ▭)
divides
. Similarly, since
and ,
it follows that divides
.
We call
, , the invariant rational functions of in
.
There is a particular case worth considering: If then
and . In this case, we obtain the invariant rational functions of at infinity (recall (▭)).
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4. Wiener–Hopf equivalence
The left Wiener–Hopf equivalence of rational matrices with respect
to a closed contour in the complex plane has been extensively
studied ([5] or [6]). Now we present the
generalization to arbitrary fields ([13]).
Definition 15
Let and be subsets of such that .
Let be two non-singular
rational matrices with no zeros and no poles in . The
matrices are said to be left Wiener–Hopf
equivalent with respect to if there exist both invertible
in and
invertible in such that
uid59
This is, in fact, an equivalence relation as it is easily seen. It
would be an equivalence relation even if no condition about the
union and intersection of and were imposed. It will be
seen later on that these conditions are natural assumptions for
the existence of unique diagonal representatives in each class.
The right Wiener–Hopf equivalence with respect to is
defined in a similar manner: There are invertible matrices in and in such that
uid60
In the following only the left Wiener–Hopf equivalence will be
considered, but, by transposition, all results hold for the right
Wiener–Hopf equivalence as well.
The aim of this section is to obtain a complete system of invariants for the
Wiener–Hopf equivalence with respect to of rational matrices,
and to obtain, if possible, a canonical form.
There is a particular case that is worth-considering: If
and , the invertible matrices in
are the biproper matrices and the invertible matrices in
are the unimodular
matrices. In this case, the left Wiener–Hopf equivalence with
respect to is the so-called
left Wiener–Hopf equivalence at infinity (see [14]). It is known that any non-singular rational matrix is left Wiener–Hopf equivalent at
infinity to a diagonal
matrix where are integers, that is, for any non-singular there exist
both a biproper matrix and a unimodular matrix such that
uid61
where are integers
uniquely determined by . They are
called the left Wiener–Hopf factorization indices at
infinity and form a
complete system of invariants for the left Wiener–Hopf
equivalence at infinity. These are the basic objects
that will produce the complete system of invariants for the left Wiener–Hopf
equivalence with respect to .
For polynomial matrices, their left Wiener–Hopf factorization
indices at infinity are the column degrees of any right equivalent
(by a unimodular matrix) column proper matrix. Namely, a polynomial
matrix is column proper if it can be written as
with non-singular, non-negative integers and
a polynomial matrix such that the degree of the th column of
smaller than , .
Let be non-singular
polynomial. There exists a unimodular matrix
such that is column proper. The column degrees of
are uniquely determined by , although is not
(see [14], p. 388[15], [16]). Since
is column proper, it can be written as
with non-singular,
and the degree of the th
column of smaller than , . Then
. Put
. Since is non-singular and
is a strictly proper matrix, is biproper, and
where .
The left Wiener–Hopf factorization indices at infinity can be
used to associate a sequence of integers with every
non-singular rational matrix and every . This is done as follows:
If
then it can always be written as
such that the global invariant rational functions of factorize in and or, equivalently, the global invariant rational functions of factorize in
(see Proposition ▭). There may be many factorizations of
this type, but it turns out (see Proposition 3.2[4] for the polynomial case)
that the left factors in all of them are right equivalent. This
means that if with the global invariant rational functions of and
factorizing in and the global invariant rational functions of and factorizing in then there is a
unimodular matrix such that . In
particular, and have the same left Wiener–Hopf
factorization indices at infinity. Thus the following definition
makes sense:
Definition 16
Let be a non-singular rational
matrix and . Let
such
that
,
the global invariant rational functions of factorize
in , and
the global invariant rational functions of factorize in
.
Then the left Wiener–Hopf factorization indices of
with respect to are defined to be the left Wiener–Hopf
factorization indices of at infinity.
In the particular case that ,
we can put and . Therefore, the left
Wiener–Hopf factorization indices of with respect to
are the left Wiener–Hopf factorization indices
of at infinity.
We prove now that the left Wiener–Hopf equivalence with respect to
can be characterized through the left Wiener–Hopf
factorization indices with respect to .
Theorem 17
Let be such that . Let , be two non-singular rational matrices with no zeros and no poles in
. The matrices and are left
Wiener–Hopf equivalent with respect to if and only if
and have the same left
Wiener–Hopf factorization indices with respect to .
Proof.-
By Proposition ▭ we can write
with the global invariant rational functions of and of factorizing in (recall that and have no zeros and no poles in ) and the global invariant rational functions of and of factorizing in
.
Assume that have the same left
Wiener–Hopf factorization indices with respect to .
By definition, and have the same left
Wiener–Hopf factorization indices with respect to
if and have the same left Wiener–Hopf
factorization indices at infinity. This means that there exist matrices
and
such that
. We have that
We aim to prove that
is invertible in and .
Since the global invariant rational functions of and factorize in , and . Moreover, is a unit in
as desired.
Now, because and and factorize in .
Therefore and
are left Wiener–Hopf equivalent with respect to .
Conversely, let and
such that
Hence,
.
Put
and . Therefore,
the global invariant rational functions of and of
factorize in , and
the global invariant rational functions of and of
factorize in
Then and are right
equivalent (see the remark previous to Definition ▭). So, there
exists such that
Thus, .
Since is biproper and is unimodular ,
have the same left Wiener–Hopf factorization
indices at infinity. Consequentially, and have the same left
Wiener–Hopf factorization indices with respect to .
In conclusion, for non-singular rational matrices with no zeros and no poles
in the left Wiener–Hopf factorization indices
with respect to form a complete system of invariants for
the left Wiener–Hopf equivalence with respect to with .
A straightforward consequence of the above theorem is the following Corollary
Corollary 18 Let be such that . Let , be non-singular
with no zeros and no poles in . Then and are left
Wiener–Hopf equivalent with respect to if and only
if for any factorizations and
satisfying the conditions
(i)–(iii) of Definition ▭, and
are left Wiener–Hopf equivalent at infinity.
Next we deal with the problem of factorizing or reducing a
rational matrix to diagonal form by Wiener–Hopf equivalence. It
will be shown that if there exists in an ideal generated by a
monic irreducible polynomial of degree equal to 1 which is not
in , then any non-singular rational matrix, with no zeros and no poles
in admits a factorization with respect to .
Afterwards, some examples will be given in which these conditions
on and are removed and factorization fails to exist.
Theorem 19
Let be such that . Assume that there are ideals in generated by linear polynomials. Let be any of them
and
a non-singular matrix with no
zeros and no poles in . There exist both invertible in
and invertible in
such that
uid72
where are integers uniquely
determined by . Moreover, they are the left
Wiener–Hopf factorization indices of with respect to .
Proof.-
The matrix can be written (see Proposition ▭) as with the global invariant rational functions of factorizing in and the global invariant rational functions of factorizing in . As are the
left Wiener–Hopf factorization indices of at infinity, there exist matrices and
such that
with . Put
and
Then
If then this matrix is invertible in
and . We only
have to prove that is invertible in . It is clear that
is in and biproper. Moreover, the global invariant rational functions of factorize in . Therefore, is invertible in
.
We prove now the uniqueness of the factorization. Assume that also
factorizes as
uid73
with integers.
Then,
uid74
The diagonal matrices have no zeros and no poles in (because
) and they are left Wiener–Hopf
equivalent with respect to . By Theorem ▭,
they have the same left Wiener–Hopf factorization indices
with respect to . Thus, for all
.
Following [5] we could call left Wiener–Hopf factorization
indices with respect to the exponents appearing
in the diagonal matrix of Theorem ▭. They are, actually, the
left Wiener–Hopf factorization indices with respect to .
Several examples follow that exhibit some remarkable features
about the results that have been proved so far. The first two examples
show that if no assumption is made on the intersection and/or
union of and then existence and/or uniqueness of diagonal
factorization may fail to exist.
Example 20
If is a polynomial matrix with zeros in
then the existence of invertible matrices and such that
with
may fail. In fact, suppose that
, . Therefore,
and . Consider
. Assume that with
a unit in and
a unit in . Thus, a nonzero
constant and which is not a
unit in .
Example 21
If then the factorization
indices with respect to may be not unique. Suppose that
, with
and , with
a unit in and
a unit in . Then can also be factorized as
with
a unit in and
a unit in .
The following example shows that if all ideals generated
by polynomials of degree equal to one are in
then a factorization as in Theorem ▭ may not exist.
Example 22
Suppose that .
Consider and
. Let
uid78
Notice that has no zeros and no poles in . We will
see that it is not possible to find invertible matrices and such that
uid79
We can write with a
unit in and . Therefore,
uid80
is invertible in
and is also left Wiener–Hopf
equivalent with respect to to the diagonal matrix
.
Assume that there exist invertible matrices
and such that
, with
integers. Notice first that is a
nonzero constant and since and is a rational function with numerator and denominator
relatively prime with , it follows that . Thus, . Let
uid81
From
we
get
uid82
uid83
uid84
uid85
As and
, we can write
and
with , and . Therefore, by (▭),
.
Hence, or . In the
same way and using (▭), or
with a polynomial.
Moreover, by (▭), must be non-negative. Hence,
. Using now (▭) and (▭) and
bearing in mind again that and
, we
conclude that and are polynomials.
We can distinguish two cases: , and . If
and , by (▭), is a polynomial
and since is proper, it is constant: .
Thus . By (▭),
. Since is
polynomial and is proper, is also constant
and then and . Consequentially,
, and . This is impossible because
is invertible.
If then , using (▭),
uid86
Notice that and
unless . Hence, if ,
which is
greater than . This cannot happen
because is proper. Thus, . In the same
way and reasoning with (▭) we get that is also
zero. This is again impossible because is invertible.
Therefore no left Wiener–Hopf factorization of with
respect to exits.
We end this section with an example where the left Wiener–Hopf factorization indices of
the matrix polynomial in the previous example are computed. Then an ideal
generated by a polynomial of degree 1 is added to and the Wiener–Hopf
factorization indices of the same matrix are obtained in two different cases.
Example 23
Let and . Consider the matrix
uid88
which has a zero at 0. It can be written as
with
uid89
where the global invariant factors of are powers of and
the global invariant factors of are relatively prime with
. Moreover, the left Wiener–Hopf factorization
indices of at infinity are 3, 1 (add the first column multiplied by
to the second column; the result is a column proper
matrix with column degrees 1 and 3). Therefore, the left
Wiener–Hopf factorization indices of with respect to
are 3, 1.
Consider now and
. There is a
unimodular matrix
,
invertible in , such that
is column proper with column degrees 3 and 2. We can write
uid90
where
is the following biproper matrix
uid91
Moreover, the denominators of its entries are powers of and
. Therefore, is
invertible in . Since
, the left Wiener–Hopf
factorization indices of with respect to are 3,
2.
If , for example, a similar procedure
shows that has as left Wiener–Hopf
factorization indices with respect to ; the same
indices as with respect to . The reason is that is not
a divisor of and so with
and as in (▭) and factorizing in .
Remark 24 It must be noticed that a procedure has been given to compute,
at least theoretically, the left Wiener–Hopf factorization indices
of any rational matrix with respect to any subset of
. In fact, given a rational matrix and ,
write with the global invariant rational
functions of factorizing in , and the global invariant
rational functions of factorizing in
(for example, using the global Smith–McMillan form of ). We need to compute the left
Wiener–Hopf factorization indices at infinity of the rational matrix .
The idea is as follows: Let be the monic least common
denominator of all the elements of . The matrix can
be written as , with
polynomial. The left Wiener–Hopf factorization
indices of at infinity are the column degrees of any
column proper matrix right equivalent to . If are the left Wiener–Hopf
factorization indices at infinity of then are the left Wiener–Hopf
factorization indices of , where (see
[4]). Free and
commercial software exists that compute such column degrees.
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5. Rosenbrock's Theorem via local rings
As said in the Introduction,
Rosenbrock's Theorem ([1]) on pole assignment by state
feedback provides, in its polynomial formulation, a complete
characterization of the relationship between the invariant factors
and the left Wiener–Hopf factorization indices at infinity of any
non-singular matrix polynomial. The precise statement of
this result is the following theorem:
Theorem 25
Let and
be
non-negative integers and monic polynomials, respectively. Then there
exists a non-singular matrix with
as invariant factors
and as left Wiener–Hopf factorization indices at infinity if
and only if the following relation holds:
uid94
Symbol appearing in (▭) is the majorization
symbol (see [17]) and it is
defined as follows: If and
are two finite sequences of real numbers and
and
are the given sequences arranged in non-increasing order then
if
uid95
with equality for .
The above Theorem ▭ can be extended to cover rational matrix
functions.
Any rational matrix can be written as where is the
monic least common denominator of all the elements of and is
polynomial. It turns out that the invariant rational functions of are the invariant
factors of divided by after canceling common factors. We
also have the following characterization of the left Wiener–
Hopf factorization indices at infinity of : these are those of plus the degree
of (see [4]). Bearing all this in mind one can easily prove (see [4])
Theorem 26
Let be integers and
irreducible rational functions,
where are monic
such that while
. Then
there exists a non-singular rational matrix with as left Wiener–Hopf factorization
indices at infinity and as global invariant rational
functions if and only if
uid97
Recall that for any rational matrix
can be factorized into two matrices (see Proposition ▭) such that the global
invariant rational functions and the left Wiener–Hopf factorization
indices at infinity of the left factor of give the invariant
rational functions and the left Wiener–Hopf factorization indices
of with respect to . Using Theorem ▭ on
the left factor of we get:
Theorem 27
Let . Let be integers and
be irreducible
rational functions such that ,
are monic polynomials factorizing in .
Then there exists a non-singular matrix with
as invariant
rational functions with respect to and as left Wiener–Hopf
factorization indices with respect to if and only if
uid99
Theorem ▭ relates the left Wiener–Hopf factorization indices with
respect to and the finite structure inside . Our last result will relate the left
Wiener–Hopf factorization indices with respect to and the structure outside ,
including that at infinity. The next Theorem is an extension of Rosenbrock's Theorem to
the point at infinity, which was proved in [4]:
Theorem 28
Let and be
integers. Then there exists a non-singular matrix
with as left
Wiener–Hopf factorization indices at infinity and as
invariant rational functions at infinity if and only if
uid101
Notice that Theorem ▭ can be
obtained from Theorem ▭ when . In
the same way, taking into account that the equivalence at infinity
is a particular case of the equivalence in
when , we can give a
more general result than that of Theorem ▭.
Specifically, necessary and sufficient conditions can be provided for
the existence of a non-singular rational matrix with prescribed left
Wiener–Hopf factorization indices with respect to and
invariant rational functions in .
Theorem 29
Let be such that . Assume that there are ideals in generated by linear polynomials and let be any of them.
Let be integers,
irreducible rational functions such that
, are monic
polynomials factorizing in and integers
such that . Then there exists a non-singular
matrix with no zeros and no poles in
with as left Wiener–Hopf factorization
indices with respect to and
as invariant rational functions in if
and only if the following condition holds:
uid103
The proof of this theorem will be given along the following two subsections.
We will use several auxiliary results that will be stated and proved when needed.
5.1. Necessity
We can give the following result for rational matrices using a similar result given in
Lemma 4.2 in [18] for matrix polynomials.
Lemma 30
Let be such that . Let be a
non-singular matrix with no zeros and no poles in with as left Wiener–Hopf factorization indices at infinity
and as left Wiener–Hopf factorization
indices with respect to . If
are the invariant rational functions of with respect to
then
uid106
It must be pointed out that may be an unordered -tuple.
Proof.- By Proposition ▭
there exist unimodular matrices such that
uid107
with , , , , units in
and factorizing in because has no poles and no zeros in .
Therefore , where
has as left Wiener–Hopf factorization indices at infinity and has as global invariant rational functions.
Let . Hence,
uid108
with units in and factorizing in . Put . Its left Wiener–Hopf factorization indices at infinity
are Lemma 2.3[4].
The matrix
has as left Wiener–Hopf factorization indices at infinity. Now if
then its invariant factors are
, and, by Lemma 4.2[18],
uid109
Therefore, (▭) follows.
5.1.1. Proof of Theorem : Necessity
If are the invariant rational functions of
in then there exist matrices invertible in
such that
uid111
We analyze first the finite structure of with respect to . If
,
we can write as follows:
uid112
with and invertible
matrices in . Thus
are the invariant rational functions of with respect to .
Let be the left Wiener–Hopf factorization indices of at
infinity. By Lemma ▭ we have
uid113
As far as the structure of at infinity is concerned, let
uid114
Then and
uid115
where and are biproper matrices. Therefore
are the invariant rational functions
of at infinity. By Theorem ▭
uid116
Let (the symmetric group of order ) be a permutation such that
and
define , . Using (▭) and
(▭) we obtain
uid117
for .
When the previous inequalities are all equalities and condition (▭)
is satisfied.
Remark 31
It has been seen in the above proof that if a matrix
has
as invariant rational functions in
then
are its invariant rational functions with respect to and
are its invariant rational
functions at infinity.
5.2. Sufficiency
Let be arbitrary elements such that . Consider the changes of indeterminate
uid120
and notice that . For , let denote
the multiplicative subset of whose elements are coprime with . For as above define
uid121
In words, if () then
uid122
In general with equality if and only if
. This shows that the restriction of to is a bijection. In addition is the restriction of to ; i.e.,
uid123
or .
In what follows we will think of as given elements of and the subindices of , and will be removed. The following are properties of (and ) that can be easily proved.
Lemma 32 Let . The following properties hold:
.
If then .
If is an irreducible polynomial then is an irreducible polynomial.
If are coprime polynomials then , are coprime polynomials.
As a consequence the map
uid129
with , is a bijection whose inverse is
uid130
where . In particular, if and (i.e. the complementary subset of in ) then
uid131
In what follows and for notational simplicity we will assume .
Lemma 33
Let where is an arbitrary element of .
If factorizes in then factorizes in .
If is a unit of then is a unit of .
Proof.- 1. Let
with constant, and .
Then . By Lemma ▭ is an irreducible polynomial (that may not be monic). If is the leading coefficient of then is monic, irreducible and . Hence factorizes in .
2. If is a unit of then it can be written as where and is a unit of . Therefore factorizes in . Since , it factorizes in (recall that we are assuming ) . So, is a unit of .
Lemma 34
Let be an arbitrary element. Then
If and then .
If then .
If then .
If and
then the matrix
Proof.- Let with .
uid140
1. Assume that and let be any element of . Therefore is a unit of and, by Lemma ▭.2, is a unit of . Moreover, is also a unit of . Hence, . Furthermore, if , it is a unit of and is a unit of .
2. If is any element of then and so
. Since may divide we have that . Hence, and . Moreover if then ,
and . Thus,
is a biproper rational function, i.e., a unit of .
3. If and is any element of
then . Since
and we obtain that . In addition, if
, which is a unit of , then and since we conclude that
is a unit of .
4. It is a consequence of 1., 2. and Remark ▭.2.
Proposition 35
Let and . If
is non-singular with
as invariant rational functions with respect to then
is a non-singular matrix with
as invariant rational functions in
where .
Proof.- Since are the invariant rational
functions of with respect to , there are such that
uid142
Notice that . Let
, which is a non-zero constant, and put
. Hence,
uid143
with
uid144
By 4 of Lemma ▭ matrices , and the Proposition follows.
Proposition 36
Let
such that . Assume that there are ideals in generated by linear polynomials and let be any of them. If
is a non-singular rational matrix with no poles and no
zeros in and as left Wiener–Hopf factorization indices with
respect to then is a non-singular rational
matrix with no poles and no zeros in and as left
Wiener–Hopf factorization indices with respect to .
Proof.- By Theorem ▭ there are matrices invertible in
and
invertible in such that
By Lemma ▭ is invertible in
and is invertible in . Moreover,
and
. Thus,
has no poles and no zeros in and
are its left Wiener–Hopf factorization indices
with respect to .
5.2.1. Proof of Theorem : Sufficiency
Let be integers,
irreducible rational functions such that
,
are monic polynomials factorizing
in and
integers such that and satisfying (▭).
Since and are coprime polynomials that factorize in
and , by Lemmas ▭ and
▭, are irreducible rational functions with
numerators and denominators polynomials factorizing in
(actually, in ) and such that each numerator divides the
next one and each denominator divides the previous one.
By (▭) and Theorem ▭ there is a matrix
with
as left Wiener–Hopf factorization indices with respect to
and
as invariant rational functions
with respect to where ,
. Notice that has no zeros and poles in
because the numerator and denominator of each rational function
factorizes
in and so it is a unit of .
Put and .
As remarked in the proof of Proposition ▭,
and .
Now so that we can apply Proposition ▭ to with the
change of indeterminate . Thus the invariant rational
functions of in are
.
On the other hand and so . Then we can apply Proposition
▭ to with so that
is a non-singular matrix with no poles and no zeros in and as left
Wiener–Hopf factorization indices with respect to
. The theorem
follows by letting .
Remark 37 Notice that when and in
Theorem ▭ we obtain Theorem ▭
().