ON SOME USES OF FACTORIZATIONS IN LINEAR SYSTEM THEORY

P.A. Fuhrmann^{∗}

∗Earl Katz Family Chair in Algebraic System Theory, Department of Mathematics, Ben-Gurion University of the Negev, Beer Sheva, Israel

Abstract. We present a short, and far from exhaustive, survey of some of the uses of factorization theory in the study of linear systems. The whole development is based on model theory, be it in the context of vectorial polynomial, rational or Hardy spaces. Invariant subspaces of operators are related to factorizations of matrix functions. The connection to system theory is via realizations based on various coprime factorizations. Among topics touched upon are the study of factorizations of rational functions and in particular the important cases of inner and all-pass functions. We survey the subject of coprime factorizations and their connection, via spectral factorizations, to various Riccati equations. In this connection we describe some geometric aspects of Wiener-Hopf factorizations. We proceed to describe connections with geometric control theory and give a Hilbert space characterization of stabilizability and detectability subspaces. Finally we describe a complete parametrization of all minimal spectral factors in the case of a coercive spectral density function.

Key Words. factorization, spectral factorization, coprime factorizations, invariant subspaces, realization theory, inner functions, Wiener-Hopf factorizations, geometric control theory, parametrization of spectral factors.

1. Introduction

Factorization theory has its roots in ancient mathematics, motivated by the study of primes and the factorization of integers into the product of primes. Rational numbers were known to the Greeks though their properties were studied mainly from the geometric viewpoint of proportions. The existence of irrational numbers stems from this source as do the Euclidean algorithm and the Bezout equation. Modular arithmetic was known in the East, e.g. the Chinese remainder theorem.

With the shift of emphasis from numbers to functions came the study zeros of functions, starting with the study of zeros of polynomials. The analogies between the ring of integers and polynomials became apparent and led to certain abstractions. The introduction of the notions of groups, rings, fields, ideals and modules led to the establishment of abstract algebra.

In this short paper we will discuss the interrelation between factorization theory and linear system theory. This connection has been a first rate example of the crossfertlization between pure and applied mathematics. Factorization results and related concepts such as coprimeness have had a great impact in the study of linear systems. However, system theory in turn has proved a great source for research problems in the area of factorization theory. Moreover, it is hard to imagine now a study of factorizations without some of the tools provided by system theory.

It is natural to seek for the underlying reasons behind this symbiotic relation. In my opinion, the reasons for this are to be found in the multifaceted way of representing linear systems. Such systems can be described externally by input/output maps, impulse responses, transferfunctions or behaviours. But they also have internal descriptions in terms of differential or difference equations, and state space models. The link connecting external and internal descriptions of a linear system is provided by realization theory. Thus external properties of a linear system are reflected by corresponding, internal, properties of realizations. Coprime factorizations are directly connected to realizations. Other properties related to factorizations have internal characterizations. Thus external properties have their counterpart in internal properties, like signature or Hamiltonian symmetry. Input/output stability is reflected in the stability of generators. The whole area of dynamic stabilization, including robustness issues, is best approached via doubly coprime factorizations.

The structure of the paper is as follows. We begin by studying ideals and invariant subspaces in spaces of functions and their representations. Set operations, which are geometric in nature are related to the arithmetic of certain factorizations. Model theory, as related to invariant subspaces and as a tool in the study of operators, is briefly described. A special case, where stability is assumed, is studied more closely. This case provides a point of contact between algebraic and analytic models.

We proceed to introduce several types of Hankel operators. The representations of their kernels and images are directly related to certain coprime factorizations, which in turn provide a key to the construction of realizations. A short study of doubly coprime factorizations (DCF) is done on two levels. We also indicate how several other classes of rational functions admit appropriate coprime factorizations. This is based on Fuhrmann and Ober [1993].

Next we describe one of the most widely applicable results on factorization of rational functions namely that of Sakhnovich [1976], which was popularized by Bart, Gohberg, Kaashoek and Van Dooren [1979]. This is a state space characterization of the existence of a factorization of a biproper rational function. The assumption is quite restrictive, but it is still the best of its kind currently available.

We mention a few results on Wiener-Hopf factorizations and the connection to invertibility of Toeplitz operators. Following that we proceed to introduce controlled and conditioned invariant subspaces, focusing on internally and externably stabilizable and detectable subspaces. We first derive an algebraic characterization of these spaces. Following that, using the fact that stability is built in, we proceed to give also an analytic characterization in terms of Hardy spaces and inner functions. These results are taken from Fuhrmann and Gombani [1995].

Inner functions are the basic component in thew description of invariant subspaces of Hardy spaces. State space characterizations of inner functions are well known, e.g. Genin et al [1983], and are given in terms of a particular solution of a homogeneous Riccati equation. We show how polynomial spectral factorization leads to an analogous characterization of inner functions. Factorizations of inner functions are related to general solutions of the same Riccati equation. This leads to the study of spectral factorizations for rational functions, positive on the imaginary axis. Extrmal spctral factors ar special casees of Wiener-Hopf factorizations. We will use these as a starting point for the parametrization of the set of all minimal spectral factors.

1 Due to limitations of space, this short survey is far from exhaustive, with the most notable omission being the approach to optimization problems via spectral and J-spectral factorizations. For some other aspects of factorization theory, in the spirit of this paper we refer to Fuhrmann [1989].

2. Polynomial and rational models

We begin by introducing polynomial models. In this paper, given a matrix function, it can be viewed alternatively as acting on column vectors by multiplication on the left or on row vectors on the right. We can pass from one case to the other by transposition, however we will find it useful to distinguish the two cases. Thus, generally, all function spaces as well as operators will have a sub(super)script, using the letters r or c, depending on whether the the space consists of row vectors or column vectors. For operators that act in a self evident way, the subscripts will be dropped.

We will denote by R^{m} the space of all mvectors with coordinates in R; R^{m} is similarly defined. Let π_{+} and π_{−} denote the projections of R^{m}((z^{−1})) the space of truncated Laurent series on R^{m}[z] and z^{−1}R^{m}[[z^{−1}]], the space of formal power series vanishing at infinity, respectively. Since

R^{m}((z^{−1})) = R^{m}[z] ⊕ z^{−1}R^{m}[[z^{−1}]]

π_{+} and π_{−} are complementary projections. In R^{m}[z] we define the shift operator by (S_{+}p)(z) = zp(z), whereas S_{−} acting in z^{−1}R^{m}[[z^{−1}]] is defined by S_{−}h = π_{−}h.

Given a nonsingular polynomial matrix D in R^{m×m}[z] we define two projections π_{D}^{c} in R^{m}[z] and π^{D} in z^{−1}R^{m}[[z^{−1}]] by

π_{D}^{c}f = Dπ_{−}D^{−1}f for f ∈ R^{m}[z]

π_{c}^{D}h = π_{−}D^{−1}π_{+}Dh for h ∈ z^{−1}R^{m}[[z^{−1}]]

and define two linear subspaces of R^{m}[z] and z^{−1}R^{m}[[z^{−1}]] by

X_{D}^{c} = Imπ_{D}^{c},

and

X_{c}^{D} = Imπ^{D}.

An element f of R^{m}[z] belongs to X_{D}^{c} if and only if π_{+}D^{−1}f = 0, i.e. if and only if D^{−1}f is a strictly proper rational vector function. The space X_{D}^{r} is analogously defined. In particular f ∈ X_{D}^{r} if and only if fD^{−1} is strictly proper. We refer to X_{D}^{r} and X_{D}^{c} as polynomial models whereas to X^{D} and X^{D} as rational models.

We turn X_{D}^{c} into an R[z]−module by defining

p · f = π_{D}^{c}pf for p ∈ R[z], f ∈ X_{D}^{c}.

Since Kerπ_{D}^{c} = DR^{m}[z] it follows that X_{D}^{c} is isomorphic to the quotient module R^{m}[z]/DR^{m}[z]. Similarly, we introduce in X_{c}^{D} a module structure by

p · h = π_{−}ph for p ∈ R[z], h ∈ X_{c}^{D}.

In X_{D}^{c} we will focus on a special map S_{D}^{c}, a generalization of the classical companion matrix, which corresponds to the action of the identity polynomial z, i.e.,

S_{D}^{c}f = π_{D}^{c}sf for f ∈ X_{D}^{c}.

Thus the module structure in X_{D}^{c} is identical to the module structure induced by S_{D}^{c} through p · f = p(S_{D}^{c})f. With this definition the study of S_{D}^{c} is identical to the study of the module structure of X_{D}^{c}. In particular the invariant subspaces of S_{D}^{c} are just the submodules of X_{D}^{c} which are characterized next. They are related to factorization of polynomial matrices.

Similarly, the module structure introduced in X_{c}^{D} is the same one induced by the shift map

S_{c}^{D}h = S_{−}h = π_{−}sh for h ∈ X_{c}^{D}.

Polynomial and rational models are closely related. Thus we have.

Proposition 1. The polynomial model X_{D}^{c} and the rational model X_{c}^{D} are isomorphic, with the isomorphism ρ_{D}^{c} : X_{c}^{D} ⟶ X_{D}^{c} given by f ↦ D^{−1}f. Moreover we have

S_{D}^{c}ρ_{D}^{c} = ρ_{D}^{c}S_{c}^{D}.

The class of polynomial and rational models is rich enough to model, up to similarity, all linear transformations acting in finite dimensional vector spaces. In fact, if A is a square matrix then S_{zI−A} is isomorphic to A. In particular this gives an easy approach to the study of canonical forms for A, via the study of the pencil zI − A.

The following is a key theorem in factorization theory. It connects S_{D}^{c}-invariant subspaces of the polynomial model X_{D}^{c}, or alternatively, S_{c}^{D}- invariant subspaces of the rational model X_{c}^{D}, with factorizations of the nonsingular polynomial matrix D. The underlying idea goes back to the work of Brodskii on factorizations of the Livsic characteristic function.

Theorem 2.

1. A subset M of X_{D}^{c} is a submodule, or equivalently an S_{D}^{c} invariant subspace, if and only if M = D_{1}X_{D_{2}}^{c} for some factorization D = D_{1}D_{2} with D_{i} ∈ R^{m×m}[z].

2. A subset M of X_{c}^{D} is a submodule, or equivalently an S_{c}^{D} invariant subspace, if and only if M = X_{c}^{D_{2}} for some factorization D = D_{1}D_{2} with D_{i} ∈ R^{m×m}[z].

We summarize now the connection between the geometry of invariant subspaces and the arithmetic of polynomial matrices.

Theorem 3. Let M_{i}, i = 1, …, s be submodules of X_{D}^{c}, having the representations M_{i} = E_{i}X_{F_{i}}, that correspond to the factorizations

D = E_{i}F_{i}.

Then the following statements are true.

(i) M_{1} ⊂ M_{2} if and only if E_{1} = E_{2}R, i.e. if and only if E_{2} is a left factor of E_{1}.

(ii) ∩_{i=1}^{s}M_{i} has the representation E_{ν}X_{F_{ν}}^{c} with E_{ν} the l.c.r.m. of the E_{i} and F_{ν} the g.c.r.d. of the F_{i}.

(iii) M_{1} + ⋯ + M_{s} has the representation E_{μ}X_{F_{μ}}^{c} with E_{μ} the g.c.l.d. of the E_{i} and F_{μ} the l.c.l.m. of all the F_{i}.

Corollary 4. Let D = E_{i}F_{i}, for i = 1, …, s. Then

(i) We have

X_{D}^{c} = E_{1}X_{F_{1}}^{c} + ⋯ + E_{s}X_{F_{s}}^{c}

if and only if the E_{i} are left coprime.

(ii) We have ∩_{i=1}^{s}E_{i}X_{F_{i}}^{c} = 0 if and only if the F_{i} are right coprime.

(iii) The decomposition

X_{D}^{c} = E_{1}X_{F_{1}}^{c} ⊕ ⋯ ⊕ E_{s}X_{F_{s}}^{c}

is a direct sum if and only if D = E_{i}F_{i} for all i, the E_{i} are left coprime and the F_{i} are right coprime.

The next result summarizes the relation between factorization and the spectral decomposition of linear maps.

Theorem 5. Let D(z) ∈ R^{n×n}[z] be nonsingular and let d(z) = detD(z) be its characteristic polynomial. Suppose d has a factorization d = e_{1} ⋯ e_{s} with the e_{i} pairwise coprime. Then D admits factorizations

D = D_{i}E_{i}

with detD_{i} = d_{i} = Π_{j≠i}e_{j}, detE_{i} = e_{i} and such that

X_{D}^{c} = D_{1}X_{E_{1}}^{c} ⊕ ⋯ ⊕ D_{s}X_{E_{s}}^{c}

Moreover

det(S_{D}|D_{i}X_{E_{i}}) = e_{i}.

Theorem 3. and Corollary 4. connect coprimeness conditions to the geometry of invariant subspaces. The next result connects coprimeness to invertibility properties of intertwining maps.

Theorem 6. Let D and \overline{D} be nonsingular polynomial matrices. Then

1. A map Z : X_{D}^{c} ⟶ Xc ; \overline{D} satisfies ZS_{D}^{c} = Sc ; \overline{D}Z

if and only if there exist polynomial matrices E and \overline{E} for which

\overline{E}D = \overline{D}E  (1)

and in terms of which

Zf = π\overline{D}\overline{E}f  for f ∈ X_{D}^{c}.  (2)

2. The map Z defined by (2) is injective if and only if D and E are right coprime.

3. The map Z defined by (2) is surjective if and only if \overline{D} and \overline{E} are left coprime.

Assuming the coprimeness conditions of the previous theorem, there exist polynomial matrices satisfying \overline{X}D − \overline{Y} E = I and \overline{D}X − \overline{E}Y = I, or in matrix form

( \begin{matrix}\overline{X} & −\overline{Y} \\ −\overline{E} & \overline{D} \end{matrix} ) ( \begin{matrix}D & Y \\ E & X \end{matrix} ) = ( \begin{matrix}I & \overline{X}Y − \overline{Y} X \\ 0 & I \end{matrix} ) .

Multiplying on the left by the inverse of the matrix on the right and modifying accordingly the definition of the matrices \overline{X}, \overline{Y}, we get the polynomial doubly coprime factorization (DCF)

( \begin{matrix}\overline{X} & −\overline{Y} \\ −\overline{E} & \overline{D} \end{matrix} ) ( \begin{matrix}D & Y \\ E & X \end{matrix} ) = ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} ) .

Doubly coprime factorizations are extremely useful. We give an example. Reversing the order of multiplication in the previous DCF, we obtain D\overline{Y} = Y \overline{D}, which indicates that the inverse of the map Z of (2) is given by Z^{−1}g = π_{D}Y g. That this is indeed true can be directly verified by a simple computation, using the Bezout equations arising from the second DCF. In the sequel we will see how DCF arise in the context of Hardy spaces.

Note that (1) can be written as \overline{D}^{−1}\overline{E} = ED^{−1}.

This opens up the possibility of applying this theorem to the analysis of realizations based on polynomial models. This is taken up next.

3. Realization theory

3.1. Shift realizations

A p×m rational matrix function can be considered as a transfer function in two ways.

We will be interested in realizations associated with rational functions having the following representations

G = V T^{−1}U + W.  (3)

where T, U, V, W are appropriately sized polynomial matrices. Each representation of the form (3), is a basis for a state space realization and corealization. This is summed up in the following theorem.

Theorem 7. Let G be a proper p × m rational function. Assume G = V T^{−1}U + W, where T, U, V, W are appropriately sized polynomial matrices.

1. Define, in the state space X_{T}^{c}, a quadruple of maps A, B, C, D, with A : X_{T}^{c} ⟶ X_{T}^{c}, B : R^{m} ⟶ X_{T}^{c}, C : X_{T}^{c} ⟶ R^{p} and D : R^{m} ⟶ R^{p} by

\{ \begin{matrix}A = S_{T} \\ Bξ = π_{T} Uξ, \\ Cf = (V T^{−1}f)_{−1} \\ D = G(∞). \end{matrix}  (4)

Then G = ( \begin{matrix}A & B \\ C & D \end{matrix} ). The realization is

reachable if and only if T and U are left coprime and observable if and only if T and V are right coprime.

2. Define, in the state space X_{T}^{r}, a system by

\{ \begin{matrix}A = S_{T}^{r} \\ ξB = ξπ_{T} V, \\ Cf = (fT^{−1}U)_{−1} \\ D = G(∞). \end{matrix}  (5)

Then G = ( \begin{matrix}D & C \\ B & A \end{matrix} ). This corealization is

reachable if and only if T and V are right coprime and observable if and only if T and U are left coprime.

Using the realization (4) and the corealization (5), as well as the isomorphism between polynomial and rational models, one can transfer these realizations to the context of rational models. We will find ample use for both representations when studying problems of geometric control theory. It should be noted that the advantage of basing a realization procedure on representations of the form (3), in contrast to one sided matrix fractions, is that it allows us to study from a functional point of view realizations that are neither reachable nor observable.

Theorem 8. Let

G = ED^{−1} = \overline{D}^{−1}\overline{E}

be a matrix fraction representation of a proper, p × m rational function.

1. In the state space X_{D}^{c} a system is defined by

\{ \begin{matrix}A = S_{D}^{c} \\ Bξ = π_{D}ξ, \\ Cp = (ED^{−1}p)_{−1} \\ D = G(∞). \end{matrix}  (6)

Then G = ( \begin{matrix}A & B \\ C & D \end{matrix} ); this realization is

reachable and it is observable if and only if E and D are right coprime.

2. In the state space Xc ; \overline{D} a system is defined by

\{ \begin{matrix}A = Sc ; \overline{D} \\ Bξ = π\overline{D}\overline{E}ξ, \\ Ch = (\overline{D}^{−1}h)_{−1} \\ D = G(∞). \end{matrix}  (7)

Then G = ( \begin{matrix}A & B \\ C & D \end{matrix} ); this realization is ob-

servable and it is reachable if and only if \overline{E} and \overline{D} are left coprime.

3. In the state space X_{c}^{D} a system is defined by

\{ \begin{matrix}A = S_{c}^{D} \\ Bξ = π_{−}D^{−1}ξ, \\ Ch = (Eh)_{−1} \\ D = G(∞). \end{matrix}  (8)

Then G = ( \begin{matrix}A & B \\ C & D \end{matrix} ); this realization is

reachable and it is observable if and only if E and D are right coprime.

4. In the state space X\overline{D} ; c a system is defined by

\{ \begin{matrix}A = S\overline{D} ; c \\ Bξ = π_{−}\overline{D}^{−1}\overline{E}ξ, \\ Ch = (h)_{−1} \\ D = G(∞). \end{matrix}  (9)

Then G = ( \begin{matrix}A & B \\ C & D \end{matrix} ); this realization is ob-

servable and it is reachable if and only if \overline{E} and \overline{D} are left coprime.

5. Under the coprimeness assumptions the realizations (6) and (7) are isomorphic. The isomorphism Z : X_{D}^{c} ⟶ Xc ; \overline{D} being given by

Zf = π\overline{D}\overline{E}f.  (10)

Similarly, the realizations (8) and (9) are isomorphic. The isomorphism \overline{Z} : X_{c}^{D} ⟶ X\overline{D} ; c being given by

\overline{Z}h = π_{−}Eh.  (11)

The polynomial coprime factorizations appearing in Theorems 6. and 8. arise most naturally in the study of Hankel operators. In fact, if G is a p × m rational matrix function, the Hankel operator H_{G} : F^{m}[z] ⟶ z^{−1}F^{p}[[z^{−1}]] is defined by

H_{G}p = π_{−}Gp.  (12)

It is easy to check that S_{−}H_{G} = H_{G}S_{+}, which shows that Ker H_{G} is a submodule of F^{m}[z] which as a linear space is of finite codimension, whereas Im H_{G} is a finite dimensional submodule of z^{−1}F^{p}[[z^{−1}]]. Thus there exist polynomial matrices D and \overline{D} for which

\{ \begin{matrix}Ker H_{G} & = & DF^{m}[z] \\ Im H_{G} & = & X\overline{D} \end{matrix}

This leads immediately to the coprime factorizations G = \overline{D}^{−1}\overline{E} = ED^{−1}. Thus a link has been established between the study of Hankel operators and that of intertwining maps. The same connection can be made also in the analytic context and is basic to the study of the Nehari problem and the commutant lifting theorem. These are central tools in H^{∞}-control.

From any minimal realization we can easily construct a basis for the model space associated with a right coprime factorization. This construction goes back to Hautus and Heymann [1978] and is extremely useful.

Theorem 9. Let G be a proper rational function of McMillan degree n. Let

G = ( \begin{matrix}A & B \\ C & D_{∞} \end{matrix} )  (13)

be a minimal realization. Then C(sI − A)^{−1} = \overline{D}^{−1}K(s) for some polynomial matrices K and \overline{D}. Defining \overline{N}(s) = K(s)B we have

G = D_{∞} + \overline{D}^{−1}\overline{N}.  (14)

A basis for Xc ; \overline{D} is given by the columns k_{i} of K

and a basis for X\overline{D} ; c is given by the columns of C(sI − A)^{−1}. In particular

X\overline{D} ; c = \{C(sI − A)^{−1}ξ | ξ ∈ R_{c}^{n}\}.  (15)

Moreover, the matrix representation of the shift realization corresponding to (14) with respect to the basis B = {h_{1}, …, h_{n}} is given by (13).

As a result we conclude that, given any polynomial matrix Q, \overline{D}^{−1}Q is strictly proper if and only if there exists a constant matrix K for which Q(z) = Φ(z)K. Similarly P D^{−1} is strictly proper if and only if there exists a constant matrix L for which P (z) = LΨ(z).

Theorem 10. Let G be a proper rational matrix

having the minimal realization G = ( \begin{matrix}A & B \\ C & D \end{matrix} ).

Then

1. If G_{1} is proper and Ker H_{G_{1}} ⊃ Ker H_{G} then G_{1} admits a realization of the form

G_{1} = ( \begin{matrix}A & B \\ C_{0} & D_{0} \end{matrix} ) .

2. If G_{1} is proper and Im H_{G_{1}} ⊂ Im H_{G} then G_{1} admits a realization of the form

G_{1} = ( \begin{matrix}A & B_{0} \\ C & D_{0} \end{matrix} ) .

Corollary. Let G = ND^{−1} have the reachable realization (A, B, C) and let G′ = MD^{−1}. Then G′ has a realization (A, B, C_{0}) for some C_{0}.

3.2. Hardy spaces

The theme of realization theory can be developed also in the analytic context of Hardy spaces. This has some disadvantages, inasmuch as stability is generally assumed, but it has the great advantage of a Hilbert space structure which is ideally suited to the study of optimization poroblems. Also it has the technical advantage that backward invariant subspaces are, simultaneously, the counterparts of both polynomial and rational models, and thus they provide a more symmetric setting.

Let L^{∞} denote the space of measurable, essentially bounded, matrix valued functions on the imaginary axis. We denote by H_{+}^{∞} the Banach space of all bounded, matrix valued, analytic functions in the open right half plane. In order not to encumber the notation, we omit the reference to the size of the matrices as this should always be clear from the context. In the case of square matrices H_{+}^{∞} is a Banach algebra. By a theorem of Fatou, these functions have nontangential boundary values a.e. on the imaginary axis. Thus H_{+}^{∞} can be identified with a closed subspace of L^{∞}. H_{+}^{2} will denote both the Hardy space of vector valued, analytic functions in the open right half plane which satisfy

∥f∥^{2} = sup_{x>0} ∫_{−∞}^{∞} ∥f(x + iy)∥^{2}dy < ∞,

as well as the subspace of L^{2} of their boundary values. Similarly we define H_{−}^{2}. As a cosequence of the Fourier-Plancherel and the Paley-Wiener theorems we have the orthogonal direct sum decomposition

L^{2} = H_{+}^{2} ⊕ H_{−}^{2}.

We denote by P_{+} and P_{−} the orthogonal projections of L^{2} onto H_{+}^{2} and H_{−}^{2} respectively.

We will use the same notation for row and column vector spaces. Usually it will be clear from the context which space is considered.

Before the introduction of Hankel operators we digress a bit on invariant subspaces of H_{+}^{2}. Since, with an eye towards continuous time problems, we are using the half planes for our definition of the H^{2} spaces, we do not have the shift operators coveniently at our disposal, and in order to keep the scope of the paper within reasonable limits we avoid introducing the translation semigroups and their Fourier transforms. This forces us to a slight departure from the usual convention.

The algebra H_{+}^{∞} can be made an algebra of operators acting on H_{+}^{2} by letting, for ψ ∈ H_{+}^{∞}, induce a multiplication map M_{ψ} : H_{+}^{2} ⟶ H_{+}^{2} which is defined by

M_{ψ}f = ψf,  f ∈ H_{+}^{2}.  (16)

In algebraic language we have introduced an H_{+}^{∞} module structure in H_{+}^{2}. The adjoint of M_{ψ} is given by

M_{ψ}^{∗}f = P_{+}ψ^{∗}f,  f ∈ H_{+}^{2}.  (17)

Both M_{ψ} and M_{ψ}^{∗} are special cases of Toeplitz operators.

A subspace M ⊂ H_{+}^{2} is called an invariant subspace if, for each ψ ∈ H_{+}^{∞} we have M_{ψ}M ⊂ M. Similarly, a subspace M ⊂ H_{+}^{2} is called a backward invariant subspace if, for each ψ ∈ H_{+}^{∞} we have M_{ψ}^{∗}M ⊂ M. Clearly backward invariant subspaces are just orthogonal complements of invariant subspaces. Invariant subspaces have been characterized by Beurling in terms of inner functions. We recall that an m × n matrix function M ∈ H_{+}^{∞} is called inner if ∥M∥_{∞} ≤ 1 and its boundary values on the imaginary axis are isometric a.e., i.e. M(it)^{∗}M(it) = I.

Theorem 11. [Beurling-Lax-Halmos] A nontrivial subspace M ⊂ H_{+}^{2} is an invariant subspace if and only if M = MH_{+}^{2} for some inner function M.

It can be shown that a nontrivial subspace M ⊂ H_{+}^{2} is an invariant subspace with a finite dimensional orthogonal complement if and only if M = MH_{+}^{2} for some rational inner function M.

The availability of Beurling’s theorem allows the arithmetization of the geometry of invariant subspaces.

Proposition 12. Let H_{+}^{2} be the Hardy space of column vector functions. Then

1. Let M, N be full invariant subspaces having the representations M = QH_{+}^{2} and N = SH_{+}^{2} respectively. Then M ⊂ N if and only if Q = ST for some inner function T.

2. Let M_{i} = Q_{i}H_{+}^{2}, i = 1, …, s be full invariant subspaces. Then

M_{1} + ⋯ + M_{s} = QH_{+}^{2}

where Q is the greatest common left inner factor of all Q_{i}.

3. Let M_{i} = Q_{i}H_{+}^{2}, i = 1, …, s be full invariant subspaces. Then

∩M_{i} = RH_{+}^{2}

where R is the least common right inner multiple of all Q_{i}.

The class of subspaces H_{c}(Q), with Q inner, is important for modelling and for the development of realization theory. Since elements ψ ∈ H_{+}^{∞} act in H_{+}^{2} by multiplication and leave QH_{+}^{2} invariant, they induce linear operators acting on H(Q) which are defined by T_{ψ,Q}f = P_{H(Q)}ψf. The adjoints are given by T_{ψ,Q}^{∗}f = P_{+}ψ^{∗}f. Clearly, if ψ ∈ H_{+}^{∞}, we have ||T_{ψ,Q}|| ≤ ||ψ||_{∞}.

We note that, for τ ≤ 0, the functions exp_{τ} (s) = e^{−τs} are all in H_{+}^{∞}. The operators T_{exp_{τ}} form a strongly continuous semigroup of operators on {MH_{+}^{2}}^{⊥}.

The results of Proposition 12. can be adapted to the context of the coinvariant subspace H_{c}(M). This provides the counterpart to Theorems 2. and 3.

Theorem 13. Let M be an inner function and H_{c}(M) = {MH_{+}^{2}}^{⊥}. Then

1. A subspace V of H_{c}(M) is invariant under all operators T_{ψ,M} if and only if V = M_{1}H_{c}(M_{2}), where M = M_{1}M_{2} is a factorization of M into the product of inner functions.

2. Let V_{i} = M_{i}H_{c}(N_{i}), i = 1, …, s be invariant subspaces of H_{c}(M) corresponding to the factorizations M = M_{i}N_{i}. Then V_{1} ⊂ V_{2} if and only if M_{1} = M_{2}R for some inner function R. Alternatively Then V_{1} ⊂ V_{2} if and only if N_{2} = RN_{1} for some inner function R.

3. We have

V_{1} + ⋯ + V_{s} = M_{γ}H_{c}(N_{γ})

where M_{γ} is the greatest common left inner factor of all M_{i}.

4. We have

∩_{i=1}^{s}V_{i} = M_{λ}H_{c}(N_{λ})

where M_{λ} is the least common right inner multiple of all M_{i}, and N_{λ} is the greatest common right inner divisor of all N_{i}.

We say that inner functions N_{i}, i = 1, …, s are mutually right coprime if, for each i, N_{i} is right coprime with the least common left inner multiple of the N_{j}, j ≠ i.

Proposition 14. The inner functions N_{i} are mutually right coprime if and only if for the inner function N defined by

H_{+}^{2}N = ∩_{i=1}^{s}H_{+}^{2}N_{i}

we have

det N = Π_{i=1}^{s} det N_{i}.

Corollary 15. Given the factorizations M = M_{i}N_{i}, we have the algebraic direct sum decomposition

H_{c}(M) = M_{1}H_{c}(N_{1}) +^{.} ⋯ +^{.} M_{s}H_{c}(N_{s})

if and only if M_{i} are mutually left coprime and N_{i} are mutually right coprime.

The following theorem contains a continuous time version of the celebrated commutant lifting theorem. In this connection see Sarason [1968] and Sz.-Nagy and Foias [1970]. The invertibility conditios are from Fuhrmann [1968].

Theorem 16. Given two square inner functions Q and \overline{Q} in H_{+}^{∞}. Then

1. A map Z : H(Q) ⟶ H(\overline{Q}) is an H_{+}^{∞}-module homomorphism. i.e. satisfies

T_{ψ,\overline{Q}}Z = ZT_{ψ,Q} for ψ ∈ H_{+}^{∞}

if and only if there exist H, \overline{H} ∈ H_{+}^{∞} satisfying

\overline{H}Q = \overline{Q}H  (18)

and in terms of which

Zf = P_{H(Q)}\overline{H}f f ∈ H(Q).  (19)

2. The map Z defined by (19) is injective if and only if Q and H are right coprime.

3. The map Z defined by (19) is surjective if and only if \overline{Q} and \overline{H} are left coprime.

4. The functions H, \overline{H} can be taken to satisfy ∥H∥_{∞} = ∥\overline{H}∥_{∞} = ∥Z∥.

Clearly, equation (18) can be rewritten as

G = \overline{Q}^{−1}\overline{H} = HQ^{−1}

which provides a factorization, referred to as the Douglas-Shapiro-Shields (DSS) factorization, of the L^{∞} function G. Not every L^{∞} function admits such a factorization, and for more on this see Fuhrmann [1981].

Given a function G ∈ L^{∞} the Hankel operator H_{G} : H_{+}^{2} ⟶ H_{−}^{2} is defined by

H_{G}f = P_{−}Gf f ∈ H_{+}^{2}.

The following result on range inclusion of Hankel operators is of use. It is taken from Fuhrmann [1975,1981].

Proposition 17. Let G, G_{1} ∈ H_{−}^{∞}. Then we have

Im H_{G_{1}} ⊂ Im H_{G}

if and only if G_{1} = GF + H for some F, H ∈ H_{+}^{∞}.

Theorem 18. Let Γ(t) be a p × m matrix function, defined on [0, ∞), whose Laplace transform, G, is strictly proper, stable and rational, i.e. in H_{+}^{2}. Then

1. A time domain minimal realization of G is given in the state space

X = span{Γ(t + τ)ξ|τ ≥ 0, ξ ∈ R^{n}} (20)

by

\{ \begin{matrix}Af & = & f′ \\ Bξ & = & Γ(t)ξ \\ Cf & = & f(0) \end{matrix}

2. Let the space X be defined by (20) and let X = LX be its Laplace transform. Then X^{⊥} is an invariant subspace of H_{+}^{2} having the representation

X^{⊥} = KH_{+}^{2}

for some rational, normalized inner function K.

3. Let the normalized, rational inner function K have the polynomial coprime factorizations

K = D^{−1}\overline{D} = \overline{E}E^{−1}

then

H_{c}(K) = {KH_{+}^{2}}^{⊥} = X_{c}^{D},

and

H_{r}(K) = {H_{+}^{2}K}^{⊥} = X_{r}^{E}.

4. With respect to the corealization (5) of K in the state space X_{r}^{E} we have

\{ \begin{matrix}(Af)(s) = sf(s) − lim_{s→∞} sf(s) \\ = S_{r}^{E}f \\ ξB = ξKπ_{−} = ξ(K(s) − I). \end{matrix}

5. The Hilbert space adjoints of the maps (A, B) defined above are, for f ∈ H_{r}(K), given by

\{ \begin{matrix}(A^{∗}f)(s) \\ = −sf(s) + lim_{σ→∞} σf(σ)K(s) \\ B^{∗}ξ = − lim_{s→∞} sf(s). \end{matrix}

6. A frequency domain minimal corealization of K is given in the state space X = LX by

\{ \begin{matrix}(Af)(s) & = & sf(s) − lim_{s→∞} sf(s) \\ ξB & = & ξ(K − I) \\ fC & = & lim_{s→∞} sf(s) \\ D & = & I. \end{matrix} (21)

Part 3. of Theorem 18. sheds light on one of the intersection points between the algebraic and analytic theories. It allows the simultaneous interpretation of the same object both as a rational model space as well as a backward invariant subspace.

Suppose now that the inner function has a factorization of the form K = K_{1}K_{2}, with K_{i} inner and K_{i}(∞) = I. This of course implies the inclusions K_{1}K_{2}H_{+}^{2} ⊂ K_{1}H_{+}^{2} and, equivalently, {K_{1}H_{+}^{2}}^{⊥} ⊂ {K_{1}K_{2}H_{+}^{2}}^{⊥}. This can be seen also from a different perspective. Let K_{1} = E_{1}\overline{D}_{1}^{−1} and K_{2} = \overline{E}_{2}D_{2}^{−1} and let E_{2}D_{1}^{−1} = \overline{D}_{1}^{−1}\overline{E}_{2}. So

\begin{aligned}K & = & E_{1}\overline{D}_{1}^{−1}\overline{E}_{2}D_{2}^{−1} = E_{1}E_{2}D_{1}^{−1}D_{2}^{−1} \\ & = & (E_{1}E_{2})(D_{2}D_{1})^{−1} \end{aligned}

Using H_{c}(K) = X_{c}^{D}, we have the inclusion X_{c}^{D_{2}D_{1}} ⊃ X_{c}^{D_{1}}. This of course is consistent with the characterization of rational model inclusions given in Theorem 2.

4. Transfer function factorization

4.1. Factorization of rational functions

Given (A_{i}, B_{i}, C_{i}), i = 1, 2, minimal realizations of G_{i} with δ(G_{i}) = dimX_{i} the McMillan degree of G_{i} then we have a realization of G_{2}G_{1} in the state space X_{1} ⊕ X_{2}. This realization may not be canonical, but it gives an upper bound on the McMillan degree of a product of rational matrices. Specifically we have the inequality δ(G_{2}G_{1}) ≤ δ(G_{1}) + δ(G_{2}).

We tackle now the inverse problem. Namely given a transfer function, when can it be factorred into the product of two transfer functions. We will assume throughout this section that the transfer function is biproper, that is proper with a proper inverse, which is equivalent to the costant term equal being invertible. If the constant term is actually equal to the identity, we will call it a normalized biproper function. In particular the inverse of a normalized biproper function is also normalized biproper.

We pass now to the basic theorem concerning factorization of transfer function due to Sakhnovich [1976], see also Bart, Gohberg, Kaashoek and van Dooren [1979].

Theorem 19. [Sakhnovich] Let G =

( \begin{matrix}A & B \\ C & I \end{matrix} ) be a biproper rational function with

the realization given in the state space X. Then a necessary and sufficient condition for G to admit a factorization G = G_{2}G_{1} with G_{i} normalized biproper rational functions is that X = M_{1} ⊕ M_{2} with M_{1} an A−invariant subspace, M_{2} an A^{×}−invariant, where A^{×} is defined by A^{×} = A − BC.

4.2. Factorization of inner functions

We proceed now to study square inner functions and their factorization properties. However in this case we are, given a normalized rational inner function K, we are interested in factorizations of the form K = K_{2}K_{1}, where the K_{i} are also normalized inner functions. It is a consequence of Beurling’s theorem that factorizations of inner functions are related to invariant subspace inclusions. However here we focus on the induced action in the model space, or backward invariant subspace H_{c}(K). To make the connection with the Sakhnovich factorization theorem we need to establish the invariance with respect to a single operator. Contrary to the case of ψ ∈ H_{+}^{∞} where the induced map on H_{c}(K) was defined by T_{ψ,K}f = P_{H_{c}(K)}ψf, the ”multiplication by s” operator is unbounded in H_{+}^{2} and hence we cannot apply the orthogonal projection P_{H_{c}(K)} to sf(s). We circumvent this difficulty, using the identification H_{c}(K) = X_{c}^{D}, where K = D^{−1}\overline{D} is a polynomial coprime factorization of K, to define the operator A : H_{c}(K) ⟶ H_{c}(K) given by Af = sf(s)−lim_{s→∞} sf(s). In fact, it was shown in Fuhrmann [1981] that H_{c}(K) = X_{c}^{D} is coinvariant, i.e. invariant under all operators M_{ψ}^{∗} if and only if it is invariant under all the maps f ↦ π_{−}pf, for an arbitrary polynomial. Obviously the last condition is equivalent to the invariance under the map S_{c}^{D}f = π_{−}sf = sf(s) − lim_{s→∞} sf(s). It is certainly not an accident that this map appears as the generator in the shift realization.

Suppose now that the inner function K has the minimal realization (21), in the state space H_{r}(K) = X_{c}^{E}. Since H_{r}(K) has a Hilbert space structure, we can compute the Hilbert space adjoints of all maps appearing in this realization. This was done in Theorem 18. But the innerness of K implies K^{∗} = K^{−1} and hence we get

fA^{∗} = −sf(s) + lim_{s→∞} sf(s) · K(s)
= −(sf(s) − lim_{s→∞} sf(s)
+ lim_{s→∞} sf(s)(K(s) − I)
= −f(A − CB)

So, if M ⊂ H_{r}(K) is a backward invariant subspace, it has to be of the form M = H_{r}(K_{1}) for a factorization K = K_{2}K_{1}. Clearly we have the orthogonal direct sum decomposition

H_{r}(K) = H_{r}(K_{1}) ⊕ H_{r}(K_{2})K_{1}

and M^{⊥} = H_{r}(K_{2})K_{1} is an A^{∗}-invariant subspace, hence also A − CB-invariant. Note that, as we deal here with row vector Hardy spaces, we have A^{×} = A − CB, differing from the definition of A^{×} in Sakhnovich’s theorem.

Before proceeding with the analysis of a state space characterization of the factorization of inner functions, we digress a bit on how spectral factorization techniques enter into the construction of rational inner functions. Note that if p is a Hurwitz polynomial and p^{∗} is defined by

p^{∗}(s) = \overline{p(−\overline{s})}, then q(s) = \frac{p^{∗}}{p} is an inner function

in the right half plane. Thus the pole structure of q determines q uniquely, up to a constant factor of modulus 1. We find it convenient to normalize the inner function by requiring that its value at infinity is equal to 1. There are other inner functions derived in the same way but of lower McMillan degree if we allow pole-zero cancellations.

In the multivariable case, the pole structure can be prescribed from either side. Moreover it can be done in two distinct but related ways. We consider the prescription of left poles. One way is to specify a stable, nonsingular, polynomial matrix to act as left denominator. The other way depends on the fact that, given such a D, there exists a unique, up to isomorphism, observable pair (A, C), with A stable, such that

X_{c}^{D} = {C(sI − A)^{−1}ξ|ξ ∈ C^{n}} .

Thus it suffices to prescribe such a pair. Naturally, in the functional form the construction depends on polynomial spectral factorization, see in this connection Callier [1985] and the further references therein. This translates to a homogeneous Riccati equation in the state space setting. This is the content of the next theorem.

Theorem 20.

1. Given an observable pair (A, C), with A sta-

ble. Then ( \begin{matrix}A & B \\ C & I \end{matrix} ) is a normalized inner

function if and only if B = −Y C^{∗} with Y a nonnegative solution of the homogeneous Riccati equation

AY + Y A^{∗} + Y C^{∗}CY = 0.  (22)

The realization ( \begin{matrix}A & B \\ C & I \end{matrix} ) is observable.

The reachable subspace is given by ImY.

2. Let D be a square, nonsingular, stable polynomial matrix. Let (A, C) be determined Then U = D^{−1}N is an inner function if and only if N is a solution of the spectral factorization problem

NN^{∗} = DD^{∗}.

The maximal McMillan degree inner function is given if we take N to be the antistable spectral factor.

As a corollary to the previous theorem we obtain the following result, giving a well known characterization of rational inner functions. See Genin, Van Dooren, Kailath, Delosme and Morf [1983], Finesso and Picci [1982] and also Fuhrmann and Ober [1993] where a slightly more general result is derived. Of course one can give a direct proof of this result based on the state space isomorphism theorem. This is the approach taken in the previously mentioned papers.

Theorem 21.

1. A function U is rational and inner in H_{+}^{∞} if and only if it has a minimal state space realization of the form

U = ( \begin{matrix}A & B \\ −B^{∗}X & I \end{matrix} )

with A stable and X the unique, positive definite solution of the homogeneous ARE

A^{∗}X + XA + XBB^{∗}X = 0

2. A function U is rational and inner in H_{+}^{∞} if and only if it has a minimal state space realization of the form

U = ( \begin{matrix}A & −Y C^{∗} \\ C & I \end{matrix} )

with A stable and Y the unique, positive definite solution of the homogeneous ARE

AY + Y A^{∗} + Y C^{∗}CY = 0

Theorem 21 establishes a connection between inner functions and the Riccati equation. On the other hand we know already the connection between factorizations of inner functions and invariant subspaces. The direct link between solutions of Riccati equations and invariant subspaces goes back to the classic paper Willems [1971]. In this paper Willems parametrizes the set of solutions of an ARE via the set of invariant subspaces of a certain related operator. For more on this theme, as well as connections to factorization theory, we refer to Finesso and Picci [1982] and Fuhrmann [1985,1989].

In view of the preceeding remarks, it is natural to develop a factorization theory for inner functions that would unify arithmetic (i.e. factorization), geometric (i.e. invariant subspaces) and dynamic (i.e. Riccati equations oriented) approaches. This leads to the following.

Theorem 22.

1. Let U be a square inner function in the right half plane, normalized so that U(∞) = I. Let

U = ( \begin{matrix}A & −Y_{+}C^{∗} \\ C & I \end{matrix} )

be a minimal realization with A stable and Y_{+} the unique, positive definite solution of the homogeneous ARE

AY + Y A^{∗} + Y C^{∗}CY = 0  (23)

Then there exists a bijective correspondence between

(a) Normalized left inner factors of U.

(b) Invariant subspaces of A.

(c) Nonnegative definite solution of the homogeneous ARE (23).

The correspondence is given by

U_{1} = ( \begin{matrix}A & −Y C^{∗} \\ C & I \end{matrix} )

where Y is a nonnegative definite solution of the homogeneous ARE (23), and V = ImY ⊂ R^{n} is A invariant.

2. Let U be a square inner function in the right half plane, normalized so that U(∞) = I. Let

U = ( \begin{matrix}A & B \\ −B^{∗}X & I \end{matrix} )

be a minimal realization with A stable and X the unique, positive definite solution of the homogeneous ARE

A^{∗}X + XA + XBB^{∗}X = 0.  (24)

Then there exists a bijective correspondence between

(a) Normalized right inner factors of U.

(b) Nonnegative definite solution of the homogeneous ARE (24).

(c) Invariant subspaces of A. The correspondence is given by

U_{2} = ( \begin{matrix}A & B \\ −B^{∗}X_{2} & I \end{matrix} )

where X_{2} is a nonnegative definite solution of the homogeneous ARE (24). and V = KerX ⊂ R^{n} is A invariant.

There is a natural partial ordering of left inner factors of a given inner function. There is another natural partial order in the set of symmetric solutions of the Riccati equation (22). It is not surprising that these partial orders are related through the following theorem. The proof is based on previous results and hence omitted.

Theorem 23.

1. Let U(s) be inner, and let U(s) = I − C(sI − A)^{−1}Y_{+}C^{∗} with Y_{+} > 0 the unique, positive definite solution of the ARE

AY + Y A^{∗} + Y C^{∗}CY = 0.  (25)

Let U_{i}(s) = I − C_{i}(sI − A)^{−1}Y_{i}C_{i}^{∗}, i = 1, 2 be two left factors of U with Y_{i} nonnegative definite solutions of the Riccati equation (25). Then U_{2} is a left inner factor of U_{1} if and only if Y_{2} ≤ Y_{1}.

2. Let U(s) be inner, and let U(s) = I − B^{∗}X(sI − A)^{−1}B with X > 0 the unique, positive definite solution of the ARE

A^{∗}X + XA + XBB^{∗}X = 0.  (26)

Let U_{i}(s) = I − B_{i}^{∗}X_{i}(sI − A)^{−1}B_{i}, i = 1, 2 be two left factors of U with X_{i} nonnegative definite solutions of the Riccati equation (26). Then U_{2} is a right inner factor of U_{1} if and only if X_{2} ≤ X_{1}.

This result has important applications in stochastic realization theory.

4.3. Coprime Factorizations Over H_{+}^{∞}

We continue our study by focusing on coprime factorizations over H_{+}^{∞}. THe motivation for this is twofold. On the one hand was the study of analytic, or rather meromorphic, continuation properties in terms of Hankel operator ranges. This was initiated by Douglas, Shapiro and Shields [1971] and carried over to the multivariable case in Fuhrmann [1975] and led to the DSS factorizations. The associated DCF are instrumental in the spectral analysis of intertwining maps, see Fuhrmann [1981]. The other driving force was stabilization theory. It turns out that stabilizing a linear system by dynamic feedback is equivalent to the solvability of a Bezout equation for the coprime factors over H_{+}^{∞}. We refer to Vidyasagar [1985] for an extensive study of the coprime factorization approach.

Two elements M, N ∈ H_{+}^{∞} are called (strongly) right coprime if there exists an H_{+}^{∞} solution to the Bezout equation \overline{V} M − \overline{U}N = I. Similarly, \overline{M}, \overline{N} ∈ H_{+}^{∞} are called (strongly) left coprime if there exists an H_{+}^{∞} solution to the Bezout equation \overline{M}V − \overline{N}U = I. Note that M, N need not have the same size, only the same number of columns. There is also a weaker notion of coprimeness in H_{+}^{∞}, see Fuhrmann [1981] for the details, but we will not discuss it here as we focus on rational functions and in this case the two notions of coprimeness coincide. In the sequel we will take coprimeness to mean the strong one.

Let G be a proper rational matrix-valued function. Then the factorization G = NM^{−1} is called a right factorization of G, if N, M are stable rational functions and M is invertible with proper inverse. If N, M are right coprime, then the factorization is called a right coprime factorization. Similarly we define a left coprime factorization G = \overline{M}^{−1}\overline{N}. The two proper stable rational block matrices

( \begin{matrix}M & U \\ N & V \end{matrix} ) , ( \begin{matrix}\overline{V} & −\overline{U} \\ −\overline{N} & \overline{M} \end{matrix} ) ,

with M (\overline{M}) having a proper inverse, form a doubly coprime factorization of the proper rational function G, if

( \begin{matrix}\overline{V} & −\overline{U} \\ −\overline{N} & \overline{M} \end{matrix} ) ( \begin{matrix}M & U \\ N & V \end{matrix} ) = ( \begin{matrix}I & 0 \\ 0 & I \end{matrix} )

and

G = NM^{−1} = \overline{M}^{−1}\overline{N}.

Doubly coprime factorizations are related to the Youla-Kucera parametrization of all stabilizing controllers. This accounts for the importance of coprime factorizations over H_{+}^{∞}.

Now, clearly, every proper rational transfer function G has a coprime factorization over H_{+}^{∞}. To see this, assume G = ED^{−1} is a polynomial right coprime factorization. Choose a stable, nonsingular polynomial matrix T such that DT^{−1} is biproper, with D, T right coprime. This can easily be done using Rosenbrock’s theorem. Thus we can write G = (ET^{−1})(DT^{−1})^{−1} = NM^{−1}, with N, M ∈ H_{+}^{∞}. This is the approach taken in Fuhrmann and Ober [1993].

Note that in the previous process we had great freedom in the choice of the denominator matrix T. This choice is severely restricted if we make additional requirements on the coprime factorizations. In fact if G ∈ H_{−}^{∞} and we require the denominators to be inner functions, we recover the DSS factorizations. If we make no restriction on G other that that it is proper and rational, but require the coprime factorizations to be normalized (NCF), i.e., with G = \overline{M}^{−1}\overline{N} = NM^{−1}, that M^{∗}M + N^{∗}N = I and \overline{M}\overline{M}^{∗} + \overline{N}\overline{N}^{∗} = I hold, the choice of T becomes essentially unique and has to be obtained via a polynomial spectral factorization procedure. To see this consider the normalized left coprime factorization case. We have N = ET^{−1} and M = DT^{−1} and the normalization condition translates into E^{∗}E +D^{∗}D = T^{∗}T. So T is obtained by polynomial spectral factorization, under the additional requirement that DT^{−1} = I. The importance of normalized coprime factorizations became clear in the work of McFarlane and Glover [1989] on robustness issues in stabilization theory. For more on this see Ober and Sefton [1991] and Georgiou and Smith [1990]. Other classes of functions that admit specially normalized coprime factorizations are the class of bounded real functions and the related class of positive real functions. All the normalizations can be given in the form

( M^{∗} N^{∗} ) J ( \begin{matrix}M \\ N \end{matrix} ) = I,

where J is an appropriate metric. For H_{−}^{∞} functions the metric is the degenerate one J_{S} =

( \begin{matrix}I & 0 \\ 0 & 0 \end{matrix} ). For the standard NCF we have J_{L} =

( \begin{matrix}I & 0 \\ 0 & I \end{matrix} ). For the bounded real case we take

J_{B} = ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ), and finally for the positive real

case the metric is chosen to be J_{P} = ( \begin{matrix}0 & I \\ I & 0 \end{matrix} ).

From a computational point of view, it is desirable to have state space formulas available for the various (normalized) coprime factorizations. A general approach to such a derivation was initiated in Khargonekar and Sontag [1982]. For normalized coprime factorizations the modified formulas were derived by Meyer and Franklin [1987] and Vidyasagar [1988]. A comprehensive study is given in Fuhrmann and Ober [1993] and the following theorem is taken from there. Some simplifying assumptions have been made in order to fit the format of this paper. Spectral factorizations are central to these derivations and are translated into the solution of various Riccati equations. Usually one Riccati equation arises for the left factorization and the other for the right one.

Theorem 24. Let G be proper and have the

minimal realization G = ( \begin{matrix}A & B \\ C & D \end{matrix} ).

1. If G ∈ H_{−}^{∞} is strictly proper, then J_{S}- normalized coprime factorizations are given by

\begin{matrix}( \begin{matrix}M_{S} \\ N_{S} \end{matrix} ) = ( \begin{matrix}A − BB^{∗}X & B \\ −B^{∗}X & I \\ C & 0 \end{matrix} ) \\ ( \overline{N}_{S} \overline{M}_{S} ) = ( \begin{matrix}A − ZC^{∗}C & −B & ZC^{∗} \\ −C & 0 & I \end{matrix} ) \end{matrix}

where X, Z are, respectively, solutions of the homogeneous Riccati equations

\begin{matrix}A^{∗}X + XA − XBB^{∗}X = 0 \\ AZ + ZA^{∗} − ZC^{∗}CZ = 0 \end{matrix}

2. For any strictly proper G, J_{L}-normalized coprime factorizations are given by

\begin{matrix}( \begin{matrix}M_{L} \\ N_{L} \end{matrix} ) = ( \begin{matrix}A − BB^{∗}X & B \\ −B^{∗}X & I \\ C & 0 \end{matrix} ) \\ ( −\overline{N}_{L} \overline{M}_{L} ) = ( \begin{matrix}A − ZC^{∗}C & B & ZC^{∗} \\ −C & 0 & I \end{matrix} ) \end{matrix}

where X, Z are, respectively, solutions of the Riccati equations

\begin{matrix}A^{∗}X + XA + C^{∗}C − XBB^{∗}X = 0 \\ AZ + ZA^{∗} + BB^{∗} − ZC^{∗}CZ = 0 \end{matrix}

3. For any strictly proper, bounded real G, J_{B}- normalized coprime factorizations are given by

( \begin{matrix}M_{B} \\ N_{B} \end{matrix} ) = ( \begin{matrix}A + BB^{∗}X & B \\ B^{∗}X & I \\ C & D \end{matrix} )

( −\overline{N}_{B} \overline{M}_{B} ) =

( \begin{matrix}A + ZC^{∗}C & B & ZC^{∗} \\ C & 0 & I \end{matrix} )

where X, Z are, respectively, solutions of the Riccati equations

\begin{matrix}A^{∗}X + XA + C^{∗}C + XBB^{∗}X = 0 \\ AZ + ZA^{∗} + BB^{∗} + ZC^{∗}CZ = 0 \end{matrix}

4. For any proper positive real G normalized so that D+D^{∗} = I, J_{P} -normalized coprime factorizations are given by

( \begin{matrix}MP \\ NP \end{matrix} ) = ( \begin{matrix}A − B(C − B∗X) & B \\ −(C − B∗X) & I \\ D∗C + DB∗X & D \end{matrix} )

( −\overline{N}P \overline{M}P ) =

( \begin{matrix}A − (B − ZC∗)C & −[BD∗ + ZC∗D] & (B − ZC∗) \\ C & −D & I \end{matrix} )

where X, Z are, respectively, solutions of the Riccati equations

(A^{∗} − C^{∗}B^{∗})X + X(A − BC)
+ C^{∗}C + XBB^{∗}X = 0
(A − BC)Z + Z(A^{∗} − C^{∗}B^{∗})
+ BB^{∗} + ZC^{∗}CZ = 0.

The derivation of doubly coprime factorizations is now easy. For the details we refer to Fuhrmann and Ober [1993].

5. Wiener-Hopf factorizations

In the analysis of stationary stochastic processes the phase function associated with a rational spectral density function is an important tool. The phase function T_{0} is defined by T_{0} = \overline{W} _{+}^{−1}W_{−}, where W_{−} is the stable, minimum phase spectral factor and W_{+} is the antistable, maximum phase spectral factor. The phase function itself is an allpass function and the previous factorization is a special case of a right Wiener-Hopf factorization, to be defined next. However, any rational all-pass function has also left and right DSS factorizations. Thus

T_{0} = K_{+}Q_{+}^{∗} = Q_{−}^{∗}K_{−},

with Q_{±}, K_{±} all inner functions. The next two theorems analyse this situation and give also characterizations of invertibility based on considerations of geometry as well based on properties of Hankel operators. The content of this section is based on Fuhrmann and Gombani [1995]. Theorem 26 is standard, see Gohberg and Feldman [1971]. Some of the results of Theorem 27 can be found in Nikolskii [1985].

We recall the definition of Toeplitz operators. As we are working with both row and column spaces we must distinguish between two types of Toeplitz operators.

Definition 25. Let G be an m × m matrix function in L^{∞}. We define the Toeplitz operator T_{G}^{c} acting on column Hardy space by

\begin{matrix}T_{G}^{c} : H_{+}^{2} ⟶ H_{+}^{2} \\ f ↦ P_{+}Gf. \end{matrix}

Similarly, we define the Toeplitz operator T_{G}^{r} acting on row Hardy space by

\begin{matrix}T_{G}^{r} : H_{+}^{2} ⟶ H_{+}^{2} \\ f ↦ P_{+}fG. \end{matrix}

In both cases G is called the symbol of the corresponding Toeplitz operator.

We say that

G = G_{−}Δ_{r}G_{+}

is a right Wiener-Hopf factorization if G_{−}^{±1} ∈ H_{−}^{∞}, G_{+}^{±1} ∈ H_{+}^{∞} and

Δ_{r}(s) = diag ((\frac{s − 1}{s + 1})^{κ_{1}} , …, (\frac{s − 1}{s + 1})^{κ_{m}})

with κ_{1} ≥ ⋯ ≥ κ_{m}. The indices κ_{1}, ⋯, κ_{m} are called the right Wiener-Hopf factorization indices. Left factorizations and left factorization indices are similarly defined.

It is well known that if G ∈ L^{∞} is continuous, and in particular if it is rational, then Wiener-Hopf factorization exist and the factorization indices, though not necessarily the factorizations, are uniquely defined.

Theorem 26. Let G ∈ L^{∞} and G = G_{−}Δ_{r}G_{+} its right Wiener-Hopf factorization. Then

1. The following statements are equivalent.

(a) The Toeplitz operator T_{G}^{c} is injective.

(b) The Toeplitz operator T_{Δ_{r}}^{c} is injective.

(c) All right Wiener-Hopf factorization indices are nonnegative.

2. The following statements are equivalent.

(a) The Toeplitz operator T_{G}^{c} is surjective.

(b) The Toeplitz operator T_{Δ_{r}}^{c} is surjective.

(c) All right Wiener-Hopf factorization indices are nonpositive.

{3. \atop (a)} The Toeplitz operator T_{G}^{c} is invertible if and only if all right Wiener-Hopf factorization indices are trivial.

(b) The Toeplitz operator T_{G}^{r} is invertible if and only if all left Wiener-Hopf factorization indices are trivial.

In the following theorem the inner functions can be taken to arise from a DSS factorization of an all-pass function, namely T_{0} = Q_{−}^{∗}K_{−} = K_{+}Q_{+}^{∗}.

Theorem 27. Let Q_{−}, Q_{+}, K_{−}, K_{+} be inner functions satisfying

Q_{−}K_{+} = K_{−}Q_{+}.

with the coprimeness conditions

\begin{aligned}K_{−} ∧_{L} Q_{−} & = & I \\ K_{+} ∧_{R} Q_{+} & = & I \end{aligned}

satisfied. Then

1. The following statements are equivalent

(a) The Toeplitz operator T_{Q_{+}K_{+}^{∗}}^{c} = T_{K_{−}^{∗}Q_{−}}^{c} is injective.

(b) The Toeplitz operator T_{K_{+}Q_{+}^{∗}}^{c} = T_{K_{−}^{∗}Q_{−}}^{c} is surjective.

(c) All right Wiener-Hopf factorization indices of Q_{+}K_{+}^{∗} = K_{−}^{∗}Q_{−} are nonnegative.

(d) We have

H_{c}(K_{−}) ∩ Q_{−}H_{+}^{2} = {0}

(e) We have

K_{−}H_{+}^{2} + Q_{−}H_{−}^{2} = L^{2}.

(f) We have

K_{−}H_{+}^{2} + H_{c}(Q_{−}) = H_{+}^{2}.

(g) We have

P_{H_{c}(K_{−})}H_{c}(Q_{−}) = H_{c}(K_{−}).

(h) All the singular values of the Hankel operator H_{K_{+}Q_{+}^{∗}}^{c} = H_{K_{−}^{∗}Q_{−}}^{c} are < 1.

2. The following statements are equivalent

(a) The Toeplitz operator T_{Q_{+}K_{+}^{∗}}^{r} = T_{K_{−}^{∗}Q_{−}}^{r} is injective.

(b) The Toeplitz operator T_{K_{+}Q_{+}^{∗}}^{r} = T_{K_{−}^{∗}Q_{−}}^{r} is surjective.

(c) All left Wiener-Hopf factorization indices of Q_{+}K_{+}^{∗} = K_{−}^{∗}Q_{−} are nonnegative.

(d) We have

H_{r}(K_{+}) ∩ H_{+}^{2}Q_{+} = {0}

(e) We have

H_{+}^{2}K_{+} + H_{−}^{2}Q_{+} = L^{2}.

(f) We have

H_{+}^{2}K_{+} + H_{r}(Q_{+}) = H_{+}^{2}.

(g) We have

P_{H_{r}(K_{+})}H_{r}(Q_{+}) = H_{r}(K_{+}).

(h) We have

3. The following statements are equivalent

(a) The Toeplitz operators T_{Q_{+}K_{+}^{∗}}^{c} = T_{K_{−}^{∗}Q_{−}}^{c} and T_{Q_{+}K_{+}^{∗}}^{r} = T_{K_{−}^{∗}Q_{−}}^{r} are both invertible.

(b) All, left and right, Wiener-Hopf factorization indices of Q_{+}K_{+}^{∗} = K_{−}^{∗}Q_{−} are trivial.

(c) We have

\{ \begin{matrix}H_{c}(K_{−}) ∩ Q_{−}H_{+}^{2} & = & \{0\} \\ H_{c}(K_{−}) + Q_{−}H_{+}^{2} & = & H_{+}^{2} \end{matrix}

(d) We have

\{ \begin{matrix}P_{H_{r}(K_{−})}H_{r}(Q_{−}) & = & H_{r}(K_{−}) \\ P_{H_{c}(Q_{−})}H_{c}(K_{−}) & = & H_{c}(Q_{−}) \end{matrix}

(e) We have

\{ \begin{matrix}K_{−}H_{−}^{2} ∩ Q_{−}H_{+}^{2} & = & \{0\} \\ K_{−}H_{−}^{2} + Q_{−}H_{+}^{2} & = & L^{2} \end{matrix}

(f) All singular values of the Hankel operators H_{Q_{+}K_{+}^{∗}}^{c} are less than 1.

dim Ker T_{Q_{+}K_{+}^{∗}}^{c} = dim{f | ∥H_{Q_{+}K_{+}^{∗}}^{c}f∥ = ∥f∥}.

6. Geometric control theory

In the analysis and synthesis of control systems, especially when using geometric methods as in Wonham [1979], it is necessary to extend the notion of an invariant subspace. Thus, given a trans-

fer function G with realization G = ( \begin{matrix}A & B \\ C & D \end{matrix} ) in

a state space X, we say that a subspace V ⊂ X is a controlled invariant subspace if given any initial condition x_{0} ∈ V there exists a control function u for which the solution to ẋ(t) = Ax(t) + Bu(t) remains in V. It is easy to check that V is controlled invariant if and only if there exists a feedback map F such that (A+BF)V ⊂ V. Thus we can introduce the dual notion. We say that a subspace W ⊂ X is conditioned invariant if there exists an output injection map H such that (A + HC)W ⊂ W. The concepts of controlled and conditioned invariant subspaces were first introduced by Basile and Marro [1969]. The following is a standard characterization. For a realization G = ( \begin{matrix}A & B \\ C & D \end{matrix} ) in the state space

X a subspace V ⊂ X is controlled invariant if and only if AV ⊂ V + ImB. Similarly, a subspace V ⊂ X is conditioned invariant if and only if A(V ∩ Ker C) ⊂ V.

It is of interest to the notions of controlled and conditioned invariant subspaces directly to the transfer function itself. This can be done using the shift realizations. The following proposition is taken from Fuhrmann and Willems [1980] and Fuhrmann [1981]. Since invariant subspaces of the shift operator are in correspondence with factorizations, it is only natural to expect that factorizations play a major role in the characterization of controlled and conditioned invariant subspaces. This is indeed the case, although the factorizations leading to the following result have been eliminated from the statement.

Proposition 28. Let G be a p×m proper rational function having the polynomial coprime factorization G = ED^{−1} = \overline{D}^{−1}\overline{E}. Then

1. With respect to the realization (8) in the state space X_{c}^{D}, a subspace V ⊂ X_{c}^{D} is controlled invariant if and only if V = π^{D}L for some submodule L ⊂ z^{−1}F^{m}[[z^{−1}]].

2. With respect to the realization (7) in the state space Xc ; \overline{D}, a subspace V ⊂ Xc ; \overline{D} is conditioned invariant if and only if V = Xc ; \overline{D} ∩ M for some submodule M ⊂ F^{p}[z].

Given a linear transformation A in X and an Ainvariant subspace V, we denote by A|_{V} the restriction of A to V. By a slight abuse of notation we will denote by A|_{X/V} the induced map, i.e. the map induced by A in the quotient space X/V. This notation extends to conditioned and controlled invariant subspaces. Thus if V ⊂ X is a controlled invariant subspace for the pair (A, B) and if F is a feedback such that (A + BF)V ⊂ V, then we use the notation A + BF |_{V} and A + BF |_{X/V} for the restricted and induced maps respectively.

Following Schumacher [1981], we say that a controlled invariant subspace V for the pair (A, B) is stabilizable, or inner stabilizable if there exists a feedback F such that V is A+BF -invariant and A + BF |_{V} is stable. Analogously, we say that a controlled invariant subspace V is outer stabilizable if there exists a feedback F such that V is A + BF -invariant and A + BF |_{X/V} is stable. Similarly we define inner antistabilizable subspaces.

There are natural dual concepts. Thus a conditioned invariant subspace V is inner detectable if there exists an injection map H such that V is A + HC-invariant and A + HC|_{V} is stable. A subspace V is outer detectable if there exists an injection map H such that V is A + HC-invariant and A + HC|_{X/V} is stable. Again, the concepts of inner and outer antidetectability are naturally defined.

A simple application of duality considerations in an inner product space X implies that V ⊂ X is an outer detectable subspace with respect to the pair (C, A) if and only if V^{⊥} is inner stabilizable with respect to the pair (A^{∗}, C^{∗}). With the previous definitions, the characterizations given in Proposition 28 can be made more specific to take into account stabilizability and detectability. To this end we define a submodule M ⊂ F^{p}[s] to be a stable submodule if M = F^{p}[s]E and E is a stable polynomial matrix. Analogously we define antistable submodules. In a similar fashion we define stable, finite dimensional, submodules of s^{−1}F^{m}[[s^{−1}]] if they are of the form L = X_{r}^{D}, with D a stable polynomial matrix. Antistable submodules require D to be antistable. Clearly stable submodules coincide with finite dimensional coinvariant subspaces of H_{+}^{2}.

The proof of the following proposition is based on results in Fuhrmann and Willems [1980] and Fuhrmann [1981].

Proposition 29. Let G be a p × m transfer function having the polynomial coprime factorizations

G = E_{r}D_{r}^{−1} = D_{l}^{−1}E_{l}.

Then

1. With respect to the corealization

\{ \begin{matrix}A = S_{D_{r}}^{r} \\ ξB = ξE_{r}π_{D_{r}}, \\ pC = (pD_{r}^{−1})_{−1} \\ D = G(∞). \end{matrix}  (27)

in the state space X_{D_{r}}^{r}, a subspace V is outer antidetectable if and only if

V = X_{D_{r}}^{r} ∩ F^{m}[s]E_{+}

for some antistable polynomial matrix E_{+}.

2. With respect to the corealization (27) in the state space X_{D_{r}}^{r}, a subspace V is outer detectable if and only if

V = X_{D_{r}}^{r} ∩ F^{m}[s]F_{−}

for some stable polynomial matrix F_{−}.

3. With respect to the corealization

\{ \begin{matrix}A = S_{r}^{D_{l}} \\ ξB = ξD_{l}^{−1}π_{−}, \\ fC = (fE_{l})_{−1} \\ D = G(∞). \end{matrix}  (28)

in the state space X_{r}^{D_{l}}, a subspace V ⊂ X_{r}^{D_{l}} is inner stabilizable if and only if there exists a stable polynomial matrix E_{−} such that

V = X_{r}^{E_{−}}π_{r}^{D_{l}}.

4. With respect to the corealization (28) in the state space X_{r}^{D_{l}}, a subspace V ⊂ X_{r}^{D_{l}} is inner antistabilizable if and only if there exists an antistable polynomial matrix E_{+} such that

V = X_{r}^{E_{+}}π_{r}^{D_{l}}.

We are ready now to state the analytic counterpart to Proposition 29.

Proposition 30. Let K be an inner function.

1. With respect to the pair A, C, defined in the state space H_{r}(K) by

\{ \begin{matrix}(Af)(s) & = & sf(s) − lim_{s→∞} sf(s) \\ fC & = & lim_{s→∞} sf(s) \end{matrix} (29)

the subspace V ⊂ H_{r}(K) is outer antidetectable if and only if

V = H_{r}(K) ∩ H_{+}^{2}Q′

for some inner function Q′.

2. Let K, Q′′ be skewprime inner functions and let KQ′′ = \overline{Q}′′K_{+}. With respect to the pair A, C, defined as in (29), the subspace V ⊂ H_{r}(K)Q′′ is outer detectable if and only if

\begin{aligned}V & = & H_{r}(K)Q′′ ∩ H_{r}(K_{+}) \\ & = & H_{r}(K)Q′′ ∩ H_{−}^{2}K_{+}. \end{aligned}

3. We have

([H_{r}(K) ∩ H_{+}^{2}Q′]Q′′)^{⊥}

= [P_{H_{r}(K)}H_{r}(Q′)]Q′′

(H_{r}(K)Q′′ ∩ H_{r}(K_{+}))^{⊥}

= P_{H_{r}(K)Q′′}(H_{r}(\overline{Q}′′)K_{+})

4. With the pair A, B, defined in the state space H_{r}(K) by

\{ \begin{matrix}(Af)(s) & = & sf(s) − lim_{s→∞} sf(s) \\ ξB & = & ξ(K − I) \end{matrix} (30)

the subspace V ⊂ H_{r}(K) is inner antistabilizable if and only if there exists an inner function Q′ such that

V = P_{H_{r}(K)}H_{r}(Q′)

5. With respect to the pair A, B, defined as in (30), the subspace V ⊂ H_{r}(K)Q′′ is inner stabilizable if and only if there exists an inner function Q′ such that

V = P_{H_{r}(K)Q′′}H_{r}(\overline{Q}′′)K_{+}.

7. Parametrization of the set of minimal spectral factors

Assume we are given a coercive, rational spectral density matrix, i.e. it is positive definite on the extended imaginary axis. Moreover, for simplicity, we assume it is normalized by Φ(∞) = I. Our aim is to parametrize the set of all minimal McMillan degree spectral factors, that is square rational matrix functions W satisfying W W^{∗} = Φ. This is best done by considering the four extremal spectral factors, W_{−}, W_{+}, \overline{W}_{−}, \overline{W}_{+}, the minimum phase stable, the maximum phase stable, the minimum phase antistable and the maximum phase antistable factors respectively. This implies the existence of four inner functions Q_{±}, K_{±} for which W_{+} = W_{−}Q_{+}, \overline{W}_{+} = \overline{W}_{−}Q_{−}, W_{−} = \overline{W}_{−}K_{−} and W_{+} = \overline{W}_{+}K_{+}. Any minimal stable spectral factor is of the form W = W_{−}Q′, where Q′ is a left factor of Q_{+} and any minimal, minimum phase spectral factor W is of the form W = \overline{W}_{−}K′, where K′ is a left factor of K_{−}. It can be shown that any minimal spectral factor W is uniquely determined by such factorizations of K_{−} and Q_{+}. This is given schematically by the following diagram.

[figure: ^{\overline{W}^{−}}; ^{\overline{W}^{+}}; W_{+}; W_{−}; W; K′; K_{+}; Q′]

Now factorizations of inner functions are in a bijective correspondence with the set of all nonnegative solutions of a corresponding homogeneous Riccati equation. Leaving out all the details, which can be found in Fuhrmann [1995], this leads to the next theorem which describes the parametrization in state space terms.

Theorem 31. Let W_{−} = ( \begin{matrix}A & B \\ C & I \end{matrix} ) be a min-

imal realization of the stable, minimum phase spectral factor of a normalized spectral function Φ. Let X be a nonnegative definite solution of the Riccati equation

X(A^{∗} − C^{∗}B^{∗}) + (A − BC)X + XC^{∗}CX = 0, and let Z be a nonnegative definite solution of the Riccati equation

A^{∗}Z + ZA + ZBB^{∗}Z = 0.

Then each of the following realizations

\begin{matrix}W1 \\ = ( \begin{matrix}A + BB∗Z & B − (I + XZ)−1X(C∗ + ZB) \\ C + B∗Z & I \end{matrix} ) \end{matrix}

and

\begin{matrix}W1 \\ = ( \begin{matrix}A + (B − XC∗)(B∗ − CX)Z(I + XZ)−1 & B − XC∗ \\ C + (B∗ − CX)Z(I + XZ)−1 & I \end{matrix} ) \end{matrix}

gives a complete parametrization of all minimal spectral factors of Φ.

The two realizations above are isomorphic and the intertwining isomorphism is given by the map I + XZ.

Acknowledgement

The author wishes to gratefully acknowledge the support of GIF under Grant No. I 184.

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