ELEMENTS OF FACTORIZATION THEORY

P. A. Fuhrmann Department of Mathematics Ben-Gurion University of the Negev Beer Sheva, Israel

DRAFT

April 23, 1995

Abstract

The paper outlines a coherent development of factorization theory in the framework of polynomial model theory. Starting from the most elementary factorizations of polynomial matrices we build up the connections to invariant subspace theory, factorizations of transfer functions, Wiener Hopf factorizations. We pass on to spectral factorizations of polynomial matrices and rational functions and the connection with the analysis of the algebraic Riccati equation. Finally we study inner/outer factorizations for a class of transfer functions and the derivation of state space formulas.

The aim throughout is to highlight the logical interconnections and the technique rather than the derivation of the most general results.

1 INTRODUCTION

Factorization theory is a typical mathematical where the interesting questions are of the inverse problem type. Once a multiplication operation is defined the question of the representation of an object as a product becomes a natural question. This becomes even more interesting when we look for a complete factorization into irreducible factors, or alternatively if certain constraints are put on the factors. Factorization problems are quite often intimately connected to structural properties and to a variety of canonical forms.

In this paper we want to outline the factorization problems most often encountered in the area of systems theory. We will start from the most elementary factorization problem, that of nonsingular polynomial matrices and study its relation to the analysis of invariant subspaces. We will see what symmetry constraints add to this problem. This will relate Hamiltonian, or parahermitian symmetry to special classes of invariant subspaces symplectic spaces.

Passing from the polynomial level to that of rational functions we analyze the factorization problem for causally invertible transfer functions. We will give a short description of Wiener-Hopf factorizations.

Our main emphasis will be on the circle of ideas centered around spectral factorizations and the algebraic Riccati quation. The central importance of this set of problems in systems theory needs no elaboration. It is a central tool in optimal control and filtering theory. Our approach is based on the spectral factorization of polynomial matrices. Finally we focus on inner/outer factorizations of rational matrix functions.

Throughout we try to outline the connection between external data, polynomial matrices, transfer functions and factorizations etc., and the internal data given in state space terms. The machinery that makes this connection clearest is the theory of polynomial models developed in Fuhrmann [1976-1988].

The history of the various factorizations is both old and rich. There is no intention of trying at a complete account or a full referencing. The following is the barest of outlines. The connection between polynomial matrices and module theory is quite old and can be found for example in McDuffee [1956] where further references to early contributors can be found. The connection between factorization of polynomial matrices and invariant subspaces derives from the work of Livsic and Brodskii in operator theory. In the context of polynomial models it can be found in Antoulas [1979] and Fuhrmann [1979].

The standard result on factorization of rational transfer functions was derived in Bart, Gohberg, Kaashoek and Van Dooren [1979], and a fuller account is given in Bart, Gohberg, and Kaashoek [1979]. Our development takes a polynomial approach and is based on Shamir and Fuhrmann [1984]. The analysis of spectral and inner-outer factorizations owes a lot, one might say it is actually based on, to Francis [1988] and Chen and Francis [1988]. In fact it arose out of trying to understand the intuition behind some of the state space formulas.

2 POLYNOMIAL MODELS

Our starting point is this basic result about free modules.

Theorem 2.1 Let R be a principal ideal domain and M a free left R−module with n basis elements. Then every R−submodule N of M is free and has at most n basis elements.

If V is a finite dimensional vector space over a field F then V [z], the space of vector polynomials is a free finitely generated module over the polynomial ring F [z]. Throughout we will assume a basis has been chosen and thus V will be identified with F^{n} and similarly V [z] with F^{n}[z]. Also we will identify F^{n}[z] and (F [z])^{n} and speak of its elements as polynomial vectors. Similarly elements of F^{m×n}[z] will be referred to as polynomial matrices. Because of the nature of the factorization results we are interested in, and for the consistency of notation, we will identify the field F with the real field R, noting that some of the results hold in greater generality.

The next theorem is the basic representation theorem for submnodules.

Theorem 2.2 A subset M of R^{n}[z] is a submodule of R^{n}[z] if and only if M = DR^{n}[z] for some D in R^{n×n}[z].

The following is the basic theorem that relates submodule inclusion to factorization.

Theorem 2.3 Let M = DR^{n}[z] and N = ER^{n}[z] then M ⊂ N if and only if D = EG for some G in R^{n×n}[z].

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[[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}]]  (1)

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

π_{D}f = Dπ_{−}D^{−1}f for f ∈ R^{m}[z]  (2)
π^{D}f = π_{−}D^{−1}π_{+}Dh for h ∈ z^{−1}R^{m}[[z^{−1}]]  (3)

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

X_{D} = Imπ_{D}.  (4)

and

X^{D} = Imπ^{D}.  (5)

An element f of R^{m}[z] belongs to X_{D} 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.

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

p · f = π_{D}pf for p ∈ R[z], f ∈ X_{D}.  (6)

Since Kerπ_{D} = DR^{m}[z] it follows that X_{D} is isomorphic to the quotient module R^{m}[z]/DR^{m}[z].

Theorem 2.4 With the previously defined module structure X_{D} is isomorphic to R^{n}[z]/DR^{n}[z].

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

S_{D}f = π_{D}zf for f ∈ D.

Thus the module structure in X_{D} is identical to the module structure induced by S_{D} through p · f = p(S_{D})f. With this definition the study of S_{D} is identical to the study of the module structure of X_{D}. In particular the invariant subspaces of S_{D} are just the submodules of X_{D} which we proceed to investigate.

The interpretation of factorization of polynomial matrices on the level of polynomial models is described next.

Theorem 2.5 A subset M of X_{D} is a submodule, or equivalently an S_{D} invariant subspace, if and only if M = D_{1}X_{D_{2}} for some factorization D = D_{1}D_{2} with D_{i} ∈ R^{n×n}[z].

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

Theorem 2.6 Let M_{i}, i = 1, …, s be submodules of X_{D}, 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}} 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}} 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 2.1 Let D = E_{i}F_{i}, for i = 1, …, s. Then

(i) We have

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

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

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

(iii) The decomposition

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

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 2.7 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}, detE_{i} = e_{i} and such that

X_{D} = D_{1}X_{E_{1}} ⊕ ⋯ ⊕ D_{s}X_{E_{s}}  (7)

Moreover

det(S_{D}|D_{i}X_{E_{i}}) = e_{i}.  (8)

Denoting by T̃ the transpose of the matrix T we define, for an element A(z) = ∑_{j=−∞}^{∞} A_{j}z^{j} of R^{m×m}((z^{−1})), Ã by

Ã(z) = ∑_{j=−∞}^{∞} Ã_{j}z^{j}.  (9)

In R^{m}((z^{−1})) × R^{m}((z^{−1})) we define a symmetric bilinear form [f, g] by

[f, g] = ∑_{j=−∞}^{∞} f̃_{j}g_{−j−1}  (10)

where f(z) = ∑_{j=−∞}^{∞} f_{j}z^{j} and g(z) = ∑_{j=−∞}^{∞} g_{j}z^{j}. It is clear that, as both f and g are truncated Laurent series, that the sum in (6) is well defined, containing only a finite number of nonzero terms. We denote by T^{∗} the adjoint of a map T relative to the bilinear form of (6), i.e. T^{∗} is the unique map that satisfies

[T f, g] = [f, T^{∗}g]  (11)

for all f, g ∈ R^{m}((z^{−1})). We use this global bilinear form to obtain a concrete representation of X_{D}^{∗} the dual space of X_{D}.

Theorem 2.8 X_{D}^{∗} The dual space of X_{D} can be identified with XD̃ under the pairing

< f, g >= [D^{−1}f, g]  (12)

for f ∈ X_{D} and g ∈ XD̃. Moreover the module structures of X_{D} and XD̃ are related through

S_{D}^{∗} = SD̃  (13)

The following result shows how the characterization of submodules of X_{D} and their relation to factorizations is reflected by duality.

Theorem 2.9 Let M ⊂ X_{D} be a submodule, represented as M = EX_{G} for some factorization D = EG into nonsingular factors. Then M^{⊥} is a submodule of XD̃ and is given by M^{⊥} = G̃XẼ.

Recall that a vector space V is symplectic if it is equipped with a nondegenerate, alternating bilinear form. A linear map H in a symplectic space V is Hamiltonian if H^{∗} = −H relative to this form. A map R in V is a symplectic if it is invertible and leaves the form invariant.

The canonical example of a symplectic space is R^{2n} with the bilinear form induced by

J = ( \begin{matrix}0 & I \\ −I & 0 \end{matrix} ) .  (14)

A Hamiltonian map H in this case is given by

H = ( \begin{matrix}A & P \\ Q & −Ã \end{matrix} )  (15)

with P, Q symmetric. R is symplectic if and only if R̃JR = R.

Contrary to inner product spaces, there are self orthogonal elements and subspaces in symplectic spaces. A subspace L of a symplectic space V is called Lagrangian if it is a maximal self orthogonal subspace.

We modify now the previous approach to duality to accomodate the study of symplectic spaces. To this end we introduce now a global alternating form on R^{m}((z^{−1})) × R^{m}((z^{−1})) in the following way. We define now a new bilinear form on R^{m}((z^{−1})) × R^{m}((z^{−1})) by

{f, g} = [τf, g] for f, g ∈ R^{m}((z^{−1}))  (16)

where τ : R^{m}((z^{−1})) → R^{m}((z^{−1})) is defined by

(τf)(z) = f(−z).  (17)

Lemma 2.1 The bilinear form defined by (16) on R^{m}((z^{−1})) × R^{m}((z^{−1})) is alternating, i.e.

{f, g} = −{g, f}.  (18)

Given a rational matrix function Φ we define its Hamiltonian or parahermitian conjugate, Φ_{∗} by

Φ_{∗}(z) = Φ̃(−z).  (19)

This implications of this symmetry were originally studied by Brockett and Rahimi [1972]. An extensive study of Hamiltonian symmetry and its associated realization theory can be found in Fuhrmann [1984].

Given a map Z : R^{m}((z^{−1})) → R^{m}((z^{−1})) we will denote by Z^{∇} the map, assuming it exists, which satisfies

{Zf, g} = {f, Z^{∇}g}  (20)

for all f, g ∈ R^{m}((z^{−1})). Since the forms [ , ], and hence also { , }, are nondegenerate the map Z^{∇}, if it exists, is unique. The map Z^{∇} will be called the Hamiltonian adjoint of Z.

Assume now that we have two nondegenerate bilinear forms on V × V^{∗} and V^{∗} × V which satisfy << x, y >>= − << y, x >> for all x ∈ V and y ∈ V^{∗}. Note that while the two forms are distinct we do not distinguish between them, as it is always clear from the context which form we use. We say x is orthogonal to y if << x, y >>= 0. Given a subset M ⊂ V we define M^{⊥} as usual by

M^{⊥} = {y ∈ V^{∗} | << x, y >>= 0 for all x ∈ M}.  (21)

Since full submodules of R^{m}[z] are of the form DR^{m}[z] with D a nonsingular polynomial matrix it is of interest to characterize its orthogonal complement. The result should be compared with Theorem 2.9.

We can use these isomorphisms to define a pairing between X_{D} and X_{(D_{∗})} by Theorem 2.10 X_{D}^{∇} The dual space of X_{D} can be identified with X_{(D_{∗})} under the pairing

<< f, g >>= {D^{−1}f, g} = [JD^{−1}f, g]  (22)

for all f ∈ X_{D} and g ∈ X_{(D_{∗})}. Moreover the module structures of X_{D} and X_{(D_{∗})} are related through

S_{D_{∗}} = −S_{(D_{∗})}.  (23)

We note at this point that if D is a nonsingular Hamiltonian symmetric polynomial matrix, i.e. if D_{∗} = D, then X_{D} with the metric of (22) is a symplectic space. Much use of this will be made later.

The pairing of elements of X_{D} and X_{D_{∗}} given by (22) allows us to compute, for a subset V of X_{D}, the annihilator V^{⊥}, i.e., the set of all g ∈ X_{D_{∗}} such that << f, g >>= 0 for all f ∈ V. Since S_{D_{∗}} = −S_{D_{∗}}, it follows that if V ⊂ X_{D} is a submodule, then so is V^{⊥} in X_{D_{∗}}. However, we know that submodules are related to factorizations of D into nonsingular factors. The annihilator of V can be concretely identified.

Theorem 2.11 Let V ⊂ X_{D} be a submodule. Then V^{⊥} is a submodule of X_{(D_{∗})}. Moreover if V = EX_{F} where D = EF is a factorization of D into nonsingular factors, then

V^{⊥} = F_{∗}X_{E_{∗}}.  (24)

If we assume D̃(−z) = D(z), i.e. D_{∗} = D, then X_{D} is a symplectic space and S_{D} a Hamiltonian map. For details see Fuhrmann [1984]. One way to obtain Hamiltonian symmetric polynomial matrices is to study those which have a factorization of the form D(z) = Ẽ(−z)E(z) or D = E_{∗}E.

Theorem 2.12 Let D = E_{∗}E. Then E_{∗}X_{E} is a Lagrangian S_{D}-invariant subspace of X_{D}.

Definition 2.1 Let P_{∗} = P be a Hamiltonian symmetric polynomial matrix. A factorization

P = N_{∗}ΣN  (25)

with a signature matrix Σ is called a Hamiltonian symmetric factorization. It is called an unmixed factorization if

σ(N) ∩ σ(N_{∗}) = ∅  (26)

or equivalently if detN and detN_{∗} are relatively prime.

The following theorem, aspecial case of an unmixed factorization of a Hamiltonian symmetric polynomial matrix, is central to the whole theory of spectral factorizations. It goes back to the work of Jacubovich [1970] and Coppel [1972]. A more accessible account is Gohberg, Lancaster and Rodman [1982] where Coppel’s proof is reproduced.

Theorem 2.13 Let Q ∈ R^{n×n}[z] be a polynomial matrix such that Q(it) > 0 for all real t. Then there exists a real polynomial matrix P such that

Q(z) = P (z)P_{∗}(z).

Moreover, P can be chosen such that all zeroes of det P lie in the open left (right) half plane.

3 REALIZATION THEORY

As usual, given a proper rational matrix G we will say a system (A, B, C, D) is a realization of G if

G = D + C(zI − A)^{−1}B.

We will use the notation G = [A, B, C, D]. We will be interested in realizations associated with rational functions having the following representations

G = V T^{−1}U + W.  (27)

Our approach to the analysis of these systems is to associate with each representation of the form (27), a state space realization in the following way. We choose X_{T} as the state space and define the triple (A, B, C), with A : X_{T} ⟶ X_{T}, B : R^{m} ⟶ X_{T}, and C : X_{T} ⟶ R^{p} by

\{ \begin{matrix}A = S_{T} \\ Bξ = π_{T} Uξ, \\ Cf = (V T^{−1}f)_{−1} \\ D = G(∞). \end{matrix}  (28)

We call this the associated realization to the polynomial matrix P.

Theorem 3.1 The system given by (28) is a realization of G = V T^{−1}U+W. This 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.

The following result as well as its dual, due to Hautus and Heymann [1978], are extremely useful.

Theorem 3.2 Let (A, C) be an observable pair, G(z) = C(zI − A)^{−1} be the corresponding state to output transfer function and let

G = T^{−1}U.

be a left coprime matrix fraction representation. Then, given any polynomial matrix N, T^{−1}N is strictly proper if and only if there exists a constant matrix K for which N(z) = U(z)K. This is equivalent to the columns of U being a basis for X_{T}.

Theorem 3.3 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}.

Let G_{1} = [A_{1}, B_{1}, C_{1}, D_{1}] and G_{2} = [A_{2}, B_{2}, C_{2}, D_{2}] be two transfer functions realized in the state spaces X_{1} and X_{2} respectively. If the number of inputs of the second system equals the number of outputs of the first we can feed those outputs to the second system. This gives rise to the series coupling and the corresponding transfer function is

G_{2}G_{1} = [( \begin{matrix}A_{1} & 0 \\ B_{2}C_{1} & A_{2} \end{matrix} ) , ( \begin{matrix}B_{1} \\ B_{2}D_{1} \end{matrix} ) , ( D_{2}C_{1} C_{2} ) , D_{2}D_{1}] .

We will use also the notation G_{2}G_{1} = [A_{2}, B_{2}, C_{2}, D_{2}] × [A_{1}, B_{1}, C_{1}, D_{1}].

Definition 3.1 We say that (A, B, C, D) acting in R^{2n} is a Hamiltonian system if, with J defined by (14), we have

\begin{aligned}ÃJ & = & −JA \\ JB & = & C̃ \\ D & = & D̃ \end{aligned}  (29)

The main result concerning Hamiltonian realizations is the following.

Theorem 3.4 (i) A transfer function G has a Hamiltonian realization if and only if it is Hamiltonian symmetric.

(ii) Two minimal Hamiltonian realizations of G are symplectically equivalent.

A modification of representation (27) leads to Hamiltonian realizations via polynomial models.

Theorem 3.5 Let G, assumed to be Hamiltonian symmetric, strictly proper and rational, have the representation

G(z) = X_{∗}(z)Q(z)^{−1}X(z) + P (z)  (30)

where X, P and Q are polynomial matrices, with Q nonsingular and Hamiltonian symmetric. Then, in the symplectic space X_{Q} the associated realization to (30) is Hamiltonian. It is minimal if and only if X and Q are left coprime.

Theorem 3.6 Let G be a p × m strictly proper rational transfer function having the representation

G(z) = V (z)T (z)^{−1}U(z) + W (z).  (31)

Let the associated realization (A, B, C), in the state space X_{T}, be defined by

\begin{aligned}A & = & S_{T} \\ Bu & = & π_{T} Uu u ∈ R^{m} \\ Cf & = & (V T^{−1}f)_{−1} for f ∈ X_{T} . \end{aligned}  (32)

Then the Hamiltonian adjoint of the realization (A, B, C) associated with this representation is (−A_{∗}, C^{∇}, B^{∇}) given by

A^{∇} = S_{T_{∗}} = −S_{T} .  (33)
B^{∇}f = (U_{∗}(T_{∗})^{−1}f)_{−1}.  (34)

and

C^{∇}u = π_{(T_{∗})}V_{∗}u.  (35)

and it is the realization associated with

G_{∗}(z) = U_{∗}(z)(T_{∗}(z))^{−1}V_{∗}(z) + W_{∗}(z).  (36)

We pass now to the polynomial characterization of conditioned invariant subspaces. We will need this in the study of the factorization problem.

Given a pair (C, A), a subspace V of the state space X is called a conditioned invariant subspace if there exists a linear transformation H such that

(A + HC)V ⊂ V.  (37)

Theorem 3.7 Let (A, B, C) be the observable realization associated with the transfer function G(z) = T (z)^{−1}U(z). Then a subspace V ⊂ X_{T} is a conditioned invariant subspace if and only if

V = E_{1}X_{F_{1}}  (38)

where T_{1} = E_{1}F_{1} is such that T_{1}^{−1}T is a bicausal isomorphism.

The following results will be needed in the study of the factorization problem.

Theorem 3.8 (i) Let M_{1}, M_{2} be two conditioned invariant subspaces of X_{T} with the representations M_{i} = E_{i}X_{F_{i}}. Then M_{1} ⊂ M_{2} if and only if E_{1} = E_{2}Y for some polynomial matrix Y.

(ii) Let M_{1}, M_{2} ⊂ X_{T} be conditioned invariant subspaces and let M_{i} = E_{i}X_{F_{i}}, i = 1, 2. Let M = M_{1} ∩ M_{2}. Then

M = X_{T} ∩ EF^{p}[z]  (39)

where E is the l.c.r.m. of E_{1}, E_{2}.

(iii) Let M_{1}, M_{2} ⊂ X_{T} be conditioned invariant subspaces, and let M_{i} = E_{i}X_{F_{i}}. Let M be the smallest conditioned invariant subspace containing both M_{1} and M_{2}. Then M = X_{T} ∩ EF^{p}[z] where E is a g.c.l.d. of E_{1} and E_{2}.

(iv) Let T and S be p × p nonsingular polynomial matrices. Then

X_{T} ∩ SF^{p}[z] = {0}  (40)

if and only if all the right Wiener-Hopf factorization indices at infinity of S^{−1}T are nonpositive.

4 INVERSE SYSTEMS

The following result is well known and has a trivial state space proof.

Theorem 4.1 Let G = [A, B, C, I] be a normalized bicausal isomorphism. Assume the realization is minimal. Then G^{−1}(z) = [A^{×}, B, −C, I] where

A^{×} = A − BC.

\underline{Proof:} It is of interest to obtain a polynomial model proof of this result. Let T^{−1}U be a left coprime factorization of π_{−}G. Then

G = I + T^{−1}U = T^{−1}(T + U) = T^{−1}D  (41)

with D defined by

D = T + U.

By our assumption T^{−1}D is a bicausal isomorphism, which implies that the polynomial models X_{D} and X_{T} contain the same elements. Since (41) implies

G^{−1} = D^{−1}T = D^{−1}(D − U) = I − D^{−1}U,

the shift realizations of G and G^{−1} have the same state space. Let (A, B, C) and (A_{1}, B_{1}, C_{1}) be the shift realizations of π_{−}G and π_{−}G^{−1} respectively, i.e.

\begin{aligned}A & = & S_{T} , \\ Bξ & = & Uξ = π_{T} Uξ \\ Cf & = & (T^{−1}f)_{−1} \end{aligned}

and

\begin{aligned}A_{1} & = & S_{D}, \\ B_{1}ξ & = & −Uξ = π_{D}T ξ \\ C_{1}f & = & (D^{−1}f)_{−1}. \end{aligned}

Clearly B_{1} = −B and, since for f ∈ X_{D}

C_{1}f = (D^{−1}f)_{−1} = (D^{−1}T T^{−1}f)_{−1} = (T^{−1}f)_{−1} = Cf,

by the fact the constant term of D^{−1}T is the identity, it follows that C_{1} = C. Finally recall that given f ∈ X_{T}

S_{T} f = zf − T ξ_{f}

for some constant vector ξ_{f}. Clearly ξ_{f} is given by

ξ_{f} = π_{+}zT^{−1}f = (T^{−1}f)_{−1}.

Now

S_{D}f = S_{T +U}f = zf − (T + U)((T + U)^{−1}f)_{−1}
= zf − (T + U)((T + U)^{−1}T T^{−1}f)_{−1}
= zf − (T + U)(T^{−1}f)_{−1}
= (zf − T (T^{−1}f)_{−1}) − U(T^{−1}f)_{−1}
= S_{T} f − U(T^{−1}f)_{−1}

or

A_{1} = A − BC = A^{×}.

Thus we have recovered the theorem modulo the trivial isomorphism given by −I.

We pass on to the study of one sided inverses. Of special interest will be the characterization of the class of transfer functions for which the singularities of the inverse system are localized in some special subsets of the complex plane.

We recall that H^{∞} is the Hardy space of all bounded analytic functions in the right half plane Π_{+}. By RH^{∞} we will denote the real subspace of all real rational functions with all poles in the open left half plane Π_{−}.

Lemma 4.1 Assume G ∈ RH^{∞} has minimal realization [A, B, C, D]. Let E = D̃(DD̃)^{−2}D and D^{+} = ED̃ = D̃(DD̃)^{−1}. Then the following statements are equivalent.

(i) G has a right inverse in RH^{∞}.

(ii) D is surjective and for some F we have

C + DF = 0  (42)

and

X_{+}(A + BF) = {0}.  (43)

(iii) D is surjective and

X_{+}(A − BD^{+}C) ⊂< A − BD^{+}C, BKerD > .  (44)

If F is such that (ii) is satisfied then a right inverse G^{+} is given by G = [A + BF, −BD^{+}, F, D^{+}].

\underline{Proof:}

(i) ⇒ (ii) Assume G has a right inverse in RH^{∞}. Then G has full row rank in Π_{+}. In particular D is surjective. Let G(z) = D + N(z)E(z)^{−1}. Then to G is associated the polynomial system matrix

P = ( \begin{matrix}E(z) & I \\ N(z) & D \end{matrix} ) .

By Theorem ?? P has full row rank for z ∈ Π_{+}. This implies the same for the polynomial system matrix

P′ = ( \begin{matrix}E(z) − D^{+}N(z) & I − D^{+}D \\ N(z) & D \end{matrix} ) = ( \begin{matrix}I & −D^{+} \\ 0 & I \end{matrix} ) ( \begin{matrix}E(z) & I \\ N(z) & D \end{matrix} ) .

Now the fact that the first row of P′ is of full rank in Π_{+} means, by a generalization of the Hautus controllability test, that the pair (S_{E−D^{+}N}, π_{E−D^{+}N} · (I − D^{+}D)) is stabilizable. On the other hand this pair is, by the Hautus Heymann [1978] polynomial characterization of state feedback, isomorphic to (A − BD^{+}C, B(I − D^{+}D)). Now the stabilizability of this pair is equivalent to

X_{+}(A − BD^{+}C) ⊂< A − BD^{+}C, BIm(I − D^{+}D) > .

and since Im(I − D^{+}D) = KerD there exists a feedback map K such that A − BD^{+}C + B(I − D^{+}D)K is stable. Putting F = −D^{+}C + (I − D^{+}D)K, then A + BF is clearly stable. Moreover

\begin{aligned}C + DF & = & C − DD^{+}C + (D − DD^{+}D)K \\ & = & (I − DD^{+})(C + DK) \end{aligned}

4 INVERSE SYSTEMS 17 (ii) ⇒ (i) Assume D is surjective and there exists an F for which (42) and (43) hold. Let D^{+} = D̃(DD̃)^{−1}. Equality (42) implies

D(F + D^{+}C) = 0

i.e. Im(F + D^{+}C) ⊂ KerD = Im(I − D^{+}D). But the last inclusion guarantees the existence of a map K for which

F + D^{+}C = (I − D^{+}D)K.

Now define G^{+}(z) by

G^{+}(z) = (I − F (zI − A − BF)^{−1}B)D^{+}.

Clearly G^{+}(z) ∈ RH^{∞}. We will show that GG^{+} = I. As

GG^{+} = D[A, B, C, I] × [A + BF, B, F, I]D^{+}

it suffices to show that

[A, B, C, I] × [A + BF, B, F, I] = I.

Using the series connection we have

\begin{matrix}[A, B, C, I] × [A + BF, B, F, I] \\ = [( \begin{matrix}A + BF & 0 \\ BF & A \end{matrix} ) , ( \begin{matrix}B \\ B \end{matrix} ) , ( F D^{+}C ) , I] \end{matrix}

Using the similarity matrix ( \begin{matrix}I & 0 \\ −I & I \end{matrix} ) the previous series connection is isomorphic to

[( \begin{matrix}A + BF & 0 \\ 0 & A \end{matrix} ) , ( \begin{matrix}B \\ 0 \end{matrix} ) , ( 0 C ) , I]

Hence GG^{+} = I.

5 TRANSFER FUNCTION FACTORIZATION

Given (A_{i}, B_{i}, C_{i}), i = 1, 2, canonical 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 factored into the product of two transfer functions. We will assume throughout this section that the transfer function is proper with costant term equal to the identity. We will call such a transfer function a normalized bicausal isomorphism. In particular the inverse of a normalized bicausal isomorphism is also a normalized bicausal isomorphism.

In much the same way as factorizations of polynomial matrices were related to invariant subspaces and controlled or conditioned invariant subspaces one expects some such relation in the case of factorizations of transfer functions. As we shall see this indeed turns out to be the case.

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

Theorem 5.1 (Bart, Gohberg, Kaashoek and Van Dooren) Let G = [A, B, C, I] be a rational normalized bicausal isomorphism with the realization in the state space X. Then a necessary and sufficient condition for G to admit a factorization

G = G_{2}G_{1}  (45)

with G_{i} normalized bicausal isomorphisms is that

X = M_{1} ⊕ M_{2}  (46)

with M_{1} an A−invariant subspace, M_{2} an A^{×}−invariant, where A^{×} is defined by

A^{×} = A − BC.  (47)

\underline{Proof:} We base our analysis of the factorization problem on polynomial models and this yields a more concrete representation of the factors.

Assume (45) is a minimal factorization such that both G_{i} are normalized bicausal isomorphisms. Let G_{1} and G_{2} have the following left coprime factorizations.

G_{1} = T_{2}^{−1}\overline{D}_{1}  (48)

and

G_{2} = \overline{T} _{1}^{−1}D_{2}.  (49)

Hence G = T_{2}^{−1}\overline{D}_{1}\overline{T} _{1}^{−1}D_{2}. Since the factorization (45) is minimal it follows that \overline{D}_{1}, \overline{T}_{1} are right coprime. Let T_{1}^{−1}D_{1} be a left coprime factorization of \overline{D}_{1}\overline{T} _{1}^{−1} then

G = T_{2}^{−1}T_{1}^{−1}D_{1}D_{2} = (T_{1}T_{2})^{−1}(D_{1}D_{2}) = T^{−1}D  (50)

and

G^{−1} = (D_{1}D_{2})^{−1}(T_{1}T_{2}).  (51)

For the shift realization of G the subspace T_{1}X_{T_{2}} of X_{T} is S_{T} −invariant. Similarly D_{1}X_{D_{2}} is S_{D}−invariant. However since

S_{D}f = S_{T} f − π_{T} D(T^{−1}f)_{−1},  (52)

D_{1}X_{D_{2}} is a conditioned invariant subspace.

We will show now the direct sum representation

X_{T} = T_{1}X_{T_{2}} ⊕ D_{1}X_{D_{2}},  (53)

and as a first step we show that

T_{1}X_{T_{2}} ∩ D_{1}X_{D_{2}} = {0}.  (54)

We know, by Theorem 3.8, that

T_{1}X_{T_{2}} ∩ D_{1}X_{D_{2}} = X_{T} ∩ ER^{p}[z]  (55)

where E is a l.c.r.m. of T_{1} and D_{1}. Thus E = T_{1}A = D_{1}B for a pair A, B of right coprime polynomial matrices. Since

T_{1}^{−1}D_{1} = AB^{−1} = \overline{D}_{1}\overline{T} _{1}^{−1}  (56)

it follows that for some unimodular matrix U, A = \overline{D}_{1}U and B = \overline{T}_{1}U. Hence

E^{−1}T = E^{−1}T_{1}T_{2} = A^{−1}T_{2} = U^{−1}(\overline{D}_{1}^{−1}T_{2}).  (57)

Thus E^{−1}T is the product of a unimodular matrix and a bicausal isomorphism and so all its right Wiener Hopf factorization indices are trivial. By Theorem ?? this implies (54).

To show equality (53) we use a dimensionality argument. From the fact that \overline{T} _{1}^{−1}D_{2} is a bicausal isomorphism it follows that deg det \overline{T}_{1} = deg det D_{2}.

Also the equality T_{1}^{−1}D_{1} = \overline{D}_{1}\overline{T} _{1}^{−1} of the two matrix fraction representations implies deg det \overline{T}_{1} = deg det T_{1}. So taking into consideration (54) we have

dim(T_{1}X_{T_{2}} + D_{1}X_{D_{2}}) = deg det T_{2} + deg det D_{2}
= deg det T_{2} + deg det \overline{T}_{1} = deg det T_{2} + deg det T_{1}
= deg det T_{1}T_{2} = dim X_{T} .

(58)

Hence equality follows.

Conversely assume now that the normalized bicausal isomorphism G is represented by the left coprime matrix fraction G = T^{−1}D such that T = T_{1}T_{2} and D = D_{1}D_{2}. Furthermore assume

X_{T} = T_{1}X_{T_{2}} ⊕ D_{1}X_{D_{2}}.  (59)

Then G has a factorization G = G_{2}G_{1} with G_{i} also normalized bicausal isomorphisms. Applying Theorem 3.8 it follows from the equality

X_{T} = T_{1}X_{T_{2}} + D_{1}X_{D_{2}}  (60)

that T_{1} and D_{1} are left coprime. Let E be the l.c.r.m. of T_{1} and D_{1}. Then, as T_{1}X_{T_{2}} ∩ D_{1}X_{D_{2}} = {0}, it follows that X_{T} ∩ ER^{p}[z] = {0} and hence the right Wiener-Hopf factorization indices are nonpositive. Let

E = T_{1}\overline{D}_{1} = D_{1}\overline{T}_{1}.  (61)

Since E is the l.c.r.m. then \overline{D}_{1}, \overline{T}_{1} are right coprime. Now

deg det E = deg det T_{1} + deg det \overline{D}_{1} =
deg det D_{1} + deg det D_{1} = deg det T_{1} + deg det T_{2}.

Here the equality deg det D_{1} = deg det T_{2} was a consequence of the direct sum representation (53). Thus necessarily E^{−1}T_{1}T_{2} has trivial right factorization indices. Thus

E^{−1}T_{1}T_{2} = UΓ  (62)

for some unimodular U and bicausal isomorphism Γ which without loss of generality we can assume normalized.

Since X_{T_{1}T_{2}} = X_{D_{1}D_{2}} also E^{−1}D_{1}D_{2} = U_{1}Γ_{1} as before. However

E^{−1}T_{1}T_{2} = (T_{1}\overline{D}_{1})^{−1}T_{1}T_{2} = \overline{D}_{1}^{−1}T_{2} = UΓ

and

E^{−1}D_{1}D_{2} = (D_{1}\overline{T}_{1})^{−1}D_{1}D_{2} = \overline{T} _{1}^{−1}D_{2} = U_{1}Γ_{1}.

The equality T_{1}\overline{D}_{1} = D_{1}\overline{T}_{1} implies

T_{1}^{−1}D_{1} = \overline{D}_{1}\overline{T} _{1}^{−1}

and hence

G = T_{2}^{−1}T_{1}^{−1}D_{1}D_{2} = (T_{2}^{−1}\overline{D}_{1})(\overline{T} _{1}^{−1}D_{2}) = Γ^{−1}U^{−1}U_{1}Γ_{1}.

This in turn implies ΓGΓ_{1}^{−1} = U^{−1}U_{1}. Thus the factorization G = G_{2}G_{1} has been established with G_{1} = Γ_{1} and G_{2} = Γ.

A remark is in order. A l.c.r.m. is only defined up to a right unimodular factor. Thus without loss of generality we can, by (62), assume E^{−1}T_{1}T_{2} is a normalized bicausal isomorphism. Thus X_{E} = X_{T_{1}T_{2}} and since

E = T_{1}\overline{D}_{1} = D_{1}\overline{T}_{1}

with T_{1}, D_{1} left coprime and \overline{T}_{1}, \overline{D}_{1} right coprime we have

X_{E} = T X_{\overline{D}_{1}} ⊕ D_{1}X_{\overline{T}_{1}}.

This means that on X_{T} we can redefine the R[z]−module structure so that both T_{1}X_{T_{2}} and D_{1}X_{D_{2}} both become submodules. Thus the invariant subspace T_{1}X_{T_{2}} and the conditioned invariant subspace D_{1}X_{D_{2}} both considered as conditioned invariant subspaces are compatible. This should come as no surprise as we know that two linearly independent controlled invariant subspaces are compatible. Hence also two conditioned invariant subspaces whose sum equals the whole state space are compatible in the sense that for some constant H both are (A + HC)−invariant, as was the case here.

We focus now on a special case of Wiener-Hopf factorizations.

Definition 5.1 Let G ∈ RH^{∞} be such that G, G^{−1} have no poles on iR. A factorization

G = G_{+}G_{−}

such that G_{−}, G_{−}^{−1} are stable and G_{+}, G_{+}^{−1} are antistable is called a canonical factorization.

Theorem 5.2 A normalized bicausal isomorphism G with minimal realization [A, B, C, I] has a canonical factorization if and only if X_{−}(A^{×}) and X_{+}(A) are complementary subspaces.

\underline{Proof:} Assume G = G_{+}G_{−} is a canonical factorization. Such a factorization is automatically minimal. Let G_{+} = T_{+}^{−1}\overline{D}_{+} and G_{−} = \overline{T} _{−}^{−1}D_{−} be a left coprime factorization.

Let \overline{D}_{+}\overline{T} _{−}^{−1} = T_{−}^{−1}D_{+} with the last one a left coprime factorization. Clearly T_{−} is stable and D_{+} antistable. Thus

G_{+}G_{−} = T_{+}^{−1}T_{−}^{−1}D_{+}D_{−} = (T_{−}T_{+})^{−1}(D_{+}D_{−})

By Theorem 5.1 we must have

X_{T_{−}T_{+}} = T_{−}X_{T_{+}} ⊕ D_{+}X_{D_{−}}  (63)

But T_{−}X_{T_{+}} = X_{+}(A) and D_{+}X_{D_{−}} = X_{−}(A^{×})

Conversely, assume X = X_{+}(A) ⊕ X_{−}(A^{×}). Thus a factorization exists. Let G = T^{−1}D. Factor T = T_{−}T_{+} and D = D_{+}D_{−} such that D_{−}, T_{−} are stable and T_{+}, D_{+} are antistable. Such factorizations exist. Also T_{−}X_{T_{+}} = X_{+}(A) and D_{+}X_{D_{−}} = X_{−}(A^{×}). By our assumption X_{T_{−}T_{+}} = T_{−}X_{T_{+}} ⊕ D_{+}X_{D_{−}} so by Theorem 5.1 this implies a factorization G = G_{+}G_{−}.

6 SPECTRAL FACTORIZATION

In this section we study the problem of spectral factorization and its relation to the algebraic Riccati equation. Our main object is to obtain state space formulas for the spectral factors.

Definition 6.1 Given a Hamiltonian symmetric normalized bicausal isomorphism Φ we define a symmetric factorization of Φ a factorization

Φ(z) = V_{∗}(z)V (z)  (64)

such that V is also a normalized bicausal isomorphism. The factorization is called minimal if δ(Φ) = 2δ(V), i.e. if there are no pole-zero cancellations between V and V_{∗}.

Theorem 6.1 Let G be a Hamiltonian symmetric transfer function having no poles or zeroes on iR and such that G(∞) > 0. Then G(z) admits a spectral factorization of the form

G(z) = G̃_{−}(−z)G_{−}(z)  (65)

with G_{−}, G_{−}^{−1} stable, This is called a spectral factorization and G_{−} is called a spectral factor.

\underline{Proof:} Let d be the characteristic polynomial of G, and let d = d_{+}d_{−} be its factorization into a stable and antistable factors. Let G = MD^{−1} be a right coprime factorization of G. Then D admits a factorization D = D_{+}D_{−} with d_{+} = detD_{+} and d_{−} = detD_{−}. Let \overline{D}_{+}^{−1}\overline{M} be a left coprime factorization of MD_{+}^{−1}. So

G(z) = MD_{+}^{−1}D_{−}^{−1} = \overline{D}_{+}^{−1}\overline{M}D_{−}^{−1}  (66)

From the fact that G is Hamiltonian we obtain

\overline{D}_{+}(z)^{−1}\overline{M}(z)D_{−}(z)^{−1} = D̃_{−}(−z)^{−1} ˜\overline{M}(−z) ˜\overline{D}_{+}^{−1}(−z)  (67)

It follows that, up to a unimodular factor which we incorporate in the polynomial matrix N, we have

\overline{D}_{+}(z) = D̃_{−}(−z)  (68)

and hence

G(z) = D̃_{−}(−z)^{−1}ND_{−}(z)^{−1}  (69)

Again, the fact that G is Hamiltonian implies Ñ(−z) = N(z).

Now the assumption that G is invertible on iR implies that N is invertible there, i.e. there are no imaginary zeroes. Since

N(it) = D̃_{−}(−it)G(it)D_{−}(it)  (70)

is positive, it follows from Theorem 2.13, that N has a spectral factorization

N(z) = Ẽ_{−}(−z)E_{−}(z).  (71)

This in turn implies the spectral factorization (65) where G_{−}(z) = E_{−}(z)D_{−}(z)^{−1}. Clearly the boundedness of G at infinity implies that of G_{−}.

We quote next the main results on the Riccati equation needed for our purposes.

Assume now that we are given the the algebraic Riccati equation (ARE)

ÃX + XA − XBB̃X + Q = 0.  (72)

With the Riccati equation we associate the Hamiltonian matrix

H = ( \begin{matrix}A & −BB̃ \\ −Q & −Ã \end{matrix} )  (73)

as well as the rational matrix function Φ(z) defined by

Φ(z) = I − B̃(zI + Ã)^{−1}Q(zI − A)^{−1}B.  (74)

Clearly Φ is an n × n causally invertible rational matrix function satisfying limΦ(z) = I, i.e. Φ is a normalized bicausal isomorphism. Moreover it is easily established that Φ is Hamiltonian symmetric.

Lemma 6.1 (i) For Φ defined by

Φ(z) = I − B̃(zI + Ã)^{−1}Q(zI − A)^{−1}B.  (75)

we have

Φ(z) = I − ( 0 B̃ ) ( \begin{matrix}zI − A & 0 \\ Q & zI + Ã \end{matrix} )^{−1} ( \begin{matrix}B \\ 0 \end{matrix} )  (76)

or

Φ = [( \begin{matrix}A & 0 \\ −Q & −Ã \end{matrix} ) , ( \begin{matrix}B \\ 0 \end{matrix} ) , ( 0 B̃ ) , I] .  (77)

(ii) For Φ as above we have

Φ(z)^{−1} = I + ( 0 B̃ ) ( \begin{matrix}zI − A & BB̃ \\ Q & zI + Ã \end{matrix} )^{−1} ( \begin{matrix}B \\ 0 \end{matrix} )  (78)

i.e.

Φ^{−1} = [H, ( \begin{matrix}B \\ 0 \end{matrix} ) , ( 0 B̃ ) , I] .  (79)

Observe also that Φ defined by Equation (74) has a realization in R^{2n}. In particular its McMillan degree δ(Φ) is bounded by 2n. Our standard assumption throughout this paper is that δ(Φ) = 2n. In particular this implies the assumption of reachability of the pair (A, B), for otherwise (zI − A)^{−1}B is not a (left) coprime factorization and we could reduce common left factors. Similarly, if Q is nonnegative definite, one can write Q = CC̃. In any such factorization (A, C) is necessarily an observable pair.

Under these assumptions we can assume without loss of generality that the spectra of A and H are disjoint. We will not dwell on this since all the details can be found in Molinari’s paper [1973b].

Observe that the realizations of Φ and Φ^{−1} given by (76) and (78) respectively are Hamiltonian realizations in the sense of Brockett and Rahimi [1972]. In this connection see also Fuhrmann [1984].

Definition 6.2 A real symmetric solution K of ARE is called an unmixed solution if

σ(A − BB̃K) ∩ σ(−Ã + KBB̃) = ∅.  (80)

The concept of unmixed solution was introduced by Shayman [1983] as a generalization of extremal solutions of the ARE studied by Willems [1971].

We are ready now to state and prove the following theorem.

Theorem 6.2 Let the matrices A, B and Q be given with (A, B) reachable and Q symmetric. Let Φ be defined by

Φ(z) = I − B̃(zI + Ã)^{−1}Q(zI − A)^{−1}B.  (81)

Let H be the Hamiltonian matrix

H = ( \begin{matrix}A & −BB̃ \\ −Q & −Ã \end{matrix} )  (82)

and let J be given by

J = ( \begin{matrix}0 & I \\ −I & 0 \end{matrix} ) .  (83)

Then there exists a bijective correspondence between the following

(i) Real symmetric solutions of the ARE.

(ii) Minimal symmetric factorizations of Φ(z).

(iii) Hamiltonian symmetric factorizations of (zI − H)J.

(iv) n−dimensional Lagrangian H−invariant subspaces of R^{2n}.

(v) Invariant subspaces of A − BB̃K for any unmixed real symmetric solution K of the ARE.

(vi) Polynomial matrix solutions N of the Polynomial Riccati Equation (PRE)

N_{∗}N = D_{∗}D + H_{∗}QH  (84)

for which ND^{−1} is an normalized bicausal isomorphism and where D, H are right coprime polynomial matrices satisfying

(zI − A)^{−1}B = H(z)D(z)^{−1}.  (85)

Theorem 6.3 Assume (A, B) is reachable. Then the following statements are equivalent:

(i) The algebraic Riccati equation has a real symmetric stabilizing solution X̂, i.e. σ(A − BB̃X̂) ⊂ Π_{−}.

(ii) The algebraic Riccati equation has a real symmetric antistabilizing solution X̌, i.e. σ(A − BB̃X̌) ⊂ Π_{+}.

(iii) For Φ defined by we have

Φ(iω) > εB̃(−iωI − Ã)^{−1}(iωI − A)^{−1}B  (86)

for some ε > 0.

(iv) For the Hamiltonian matrix H we have σ(H) ∩ iR = ∅.

Clearly the reachability of the pair (A, B) is not necessary for the existence of a real symmetric stabilizing solution. The following result makes this precise.

Theorem 6.4 The algebraic Riccati equation has a real symmetric stabilizing solution if and only if the following two conditions hold

(i) The pair (A, B) is stabilizable.

(ii) For the Hamiltonian matrix H we have σ(H) ∩ iR = ∅.

Next we obtain a state space representation of the spectral factors.

Theorem 6.5 Let G = [A, B, C, D] be a Hamiltonian symmetric transfer function having no poles or zeroes on iR and such that G(∞) > 0. and let

G(z) = D + H_{+}(z) + H_{−}(z)  (87)

be such that H_{+} and H_{−} are strictly proper and stable and antistable respectively. Let H = [A_{−}, B_{−}, C_{−}] with the realization assumed minimal. Then the right spectral factor of G is given by G_{−} = [A_{−}, B_{−}, D^{− \frac{1}{2}} (C_{−} − B̃_{−}X), D^{\frac{1}{2}} ] where D^{\frac{1}{2}} is the positive square root of D and X is the unique stabilizing solution of the algebraic Riccati equation

X(A_{−}−B_{−}D^{−1}C_{−})+(Ã_{−}−C̃_{−}D^{−1}B̃_{−})X+XB_{−}D^{−1}B̃_{−}X+C̃_{−}D^{−1}C_{−} = 0, (88)

\underline{Proof:} Let G(∞) = D. Let d be the characteristic polynomial of G, and let d = d_{+}d_{−} be its factorization into a stable and antistable factors respectively. Let G(z) = D + H_{+}(z) + H_{−}(z) be a partial fraction decomposition of G. So H_{−} and H_{+} are strictly proper and have their poles in the left and right half planes respectively. As G is Hamiltonian clearly D̃ = D and H_{+}(z) = H̃_{−}(−z). Let H_{−} = N_{−}D_{−}^{−1} be a right coprime factorization of H_{−}. Associate with H_{−} a minimal realization

H_{−}(z) = C_{−}(zI − A_{−})^{−1}B_{−}.  (89)

Clearly

H_{+}(z) = −B̃_{−}(zI + Ã_{−})^{−1}C̃_{−}.  (90)

Now

\begin{aligned}G(z) & = & D + D̃_{−}^{−1}Ñ_{−} + N_{−}D_{−}^{−1} \\ & = & D̃_{−}^{−1}\{D̃_{−}DD_{−} + Ñ_{−}D_{−} + D̃_{−}N_{−}\}D_{−}^{−1} \\ & = & D̃_{−}^{−1}ND_{−}^{−1} \end{aligned}  (91)

Obviously N is Hamiltonian symmetric.

Now the assumption that G is invertible on iR implies that N is invertible there too, i.e. there are no imaginary zeroes. Since

N(it) = D̃_{−}(−it)G(it)D_{−}(it)  (92)

is positive, it follows from Theorem 2.13, that N has a spectral factorization

N(z) = Ẽ_{−}(−z)E_{−}(z).  (93)

This in turn implies the spectral factorization (65) where G_{−}(z) = E_{−}(z)D_{−}(z)^{−1}. Clearly the boundedness of G at infinity implies that of G_{−}.

Now G(z) is the parallel connection of H_{+}(z) and H_{−}(z) and has constant term D, so it has a realization

[( \begin{matrix}A_{−} & 0 \\ 0 & −Ã_{−} \end{matrix} ) , ( B_{−} ; C̃_{−} ) , ( C_{−} −B̃_{−} ) , D] .  (94)

On the other hand we proved G has a factorization

G(z) = G_{+}(z)G_{−}(z) = G̃_{−}(−z)G_{−}(z)  (95)

where G_{−}(z) = E_{−}(z)D_{−}(z)^{−1}. Comparing this with H_{−}(z) = N_{−}(z)D_{−}(z)^{−1} it follows that G_{−} has a realization [A_{−}, B_{−}, C_{0}, D^{\frac{1}{2}} ] where C_{0} has to be determined. This realization implies the realization [−Ã_{−}, C̃_{0}, −B̃_{−}, D^{\frac{1}{2}} ] for G̃_{(} − z). Hence G as a series coupling of G_{−} and G̃_{(} − z) has a realization

[( \begin{matrix}A_{−} & 0 \\ C̃_{0}C_{0} & −Ã_{−} \end{matrix} ) , ( \begin{matrix}B_{−} \\ C̃_{0}D^{\frac{1}{2}} \end{matrix} ) , ( D \frac{1}{2} C_{0} −B̃_{−} ) , D] .  (96)

As both realizations (94) and (96) are canonical they are, by the state space isomorphism theorem, isomorphic. Based on the study of skew primeness, we

will look for a map of the form ( \begin{matrix}I & 0 \\ X & I \end{matrix} ), with X symmetric, intertwining

the two realizations. The intertwining relation reduces to the two matrix equations Thus we have to have

( \begin{matrix}I & 0 \\ X & I \end{matrix} ) ( \begin{matrix}A_{−} & 0 \\ C̃_{0}C_{0} & −Ã_{−} \end{matrix} ) = ( \begin{matrix}A_{−} & 0 \\ 0 & −Ã_{−} \end{matrix} ) ( \begin{matrix}I & 0 \\ X & I \end{matrix} ) ,  (97)

as well as

( C_{−} −B̃_{−} ) ( \begin{matrix}I & 0 \\ X & I \end{matrix} ) = ( D \frac{1}{2} C_{0} −B̃_{−} ) .  (98)

The third relation follows from the previous one by symmetry.

Thus we must have

Ã_{−}X + XA_{−} = −C̃_{0}C_{0}  (99)

and

C_{−} − B̃_{−}X = D^{\frac{1}{2}} C_{0}  (100)

Thus C_{0} = D^{− \frac{1}{2}} (C_{−} − B̃_{−}X) and substituting this back into equation (99) we obtain the algebraic Riccati equation

X(A_{−}−B_{−}D^{−1}C_{−})+(Ã_{−}−C̃_{−}D^{−1}B̃_{−})X+XB_{−}D^{−1}B̃_{−}X+C̃_{−}D^{−1}C_{−} = 0,

(101)

i.e. X must be a solution to the ARE that corresponds to the Hamiltonian matrix

H = ( \begin{matrix}A_{−} − B_{−}D^{−1}C_{−} & −B_{−}D^{−1}B̃_{−} \\ C̃_{−}D^{−1}C_{−} & −(Ã_{−} − C̃_{−}D^{−1}B̃_{−}) \end{matrix} )  (102)

Now, from the realization (94) it follows that G^{−1} is realized by

\begin{matrix}[( \begin{matrix}A_{−} − B_{−}D^{−1}C_{−} & −B_{−}D^{−1}B̃_{−} \\ C̃_{−}D^{−1}C_{−} & −(Ã_{−} − C̃_{−}D^{−1}B̃_{−}) \end{matrix} ) , ( \begin{matrix}B_{−}D^{−1} \\ −C̃_{−}D^{−1} \end{matrix} ) , \\ ( D^{−1}C_{−} −D^{−1}B̃_{−} ) , D^{−1}] \end{matrix} (103)

However G^{−1}(z) = D_{−}(Ẽ_{−}E_{−})^{−1}D̃_{−} has a Hamiltonian realization in X_{Ẽ_{−}E_{−}} with generator S_{Ẽ_{−}E_{−}}, so the two realizations are symplectically equivalent. Since the last map has a Lagrangian invariant subspace, namely Ẽ_{−}X_{E_{−}}, so has the matrix H. Hence by Theorem 6.2, the corresponding Riccati equation is solvable. Since G_{−} = E_{−}D_{−}^{−1} is a spectral factor both G and G^{−1} are stable.

Now as G_{−} has the realization [A_{−}, B_{−}, C_{0}, D^{\frac{1}{2}} ] it follows that

G_{−}^{−1} = [A_{−} − B_{−}D^{− \frac{1}{2}} C_{0}, B_{−}D^{− \frac{1}{2}} , D^{− \frac{1}{2}} C_{0}, D^{− \frac{1}{2}} ].

The stability of G_{−}^{−1} means that A_{−} − B_{−}D^{− \frac{1}{2}} C_{0} is also stable. However it is easily computed that

\begin{matrix}( \begin{matrix}A_{−} − B_{−}D^{−1}C_{−} & −B_{−}D^{−1}B̃_{−} \\ C̃_{−}D^{−1}C_{−} & −(Ã_{−} − C̃_{−}D^{−1}B̃_{−}) \end{matrix} ) & ( \begin{matrix}I & 0 \\ X & I \end{matrix} ) \\ = ( \begin{matrix}I & 0 \\ X & I \end{matrix} ) ( \begin{matrix}A_{−} − B_{−}D^{−1}C_{0} & −B_{−}D^{−1}B̃_{−} \\ 0 & −(Ã_{−} − C̃_{0}D^{−1}B̃_{−}) \end{matrix} ) \end{matrix}

so that ( \begin{matrix}I \\ X \end{matrix} ) is a basis matrix for X_{−}(H). In fact if ( \begin{matrix}U \\ V \end{matrix} ) is any basis

matrix for X_{−}(H) then necessarily U is invertible and X = V U^{−1}.

An obvious way to construct Hamiltonian symmetric transfer functions is to study transfer functions of the form G̃(−z)G(z) where G is an arbitrary real transfer function. The next theorem tackles spectral factorizations of functions given in this form.

Theorem 6.6 Let G ∈ RH^{∞} be an injective proper transfer function with realization [A, B, C, D]. Then G_{∗}G has a spectral factorization G_{−∗}G_{−}, with the spectral factor G_{−} realized by

[A, B, \underline{D}^{− \frac{1}{2}} (D̃C − B̃X), \underline{D}^{\frac{1}{2}} ]  (104)

and X is the stabilizing solution of the algebraic Riccati equation corresponding to the Hamiltonian matrix

( \begin{matrix}A − B\underline{D}^{−1}D̃C & −B\underline{D}^{−1}B̃ \\ −(C̃C − C̃D\underline{D}^{−1}D̃C) & −(Ã − C̃D\underline{D}^{−1}B̃) \end{matrix} )  (105)

\underline{Proof:} Since we assume G is left invertible it follows that G(∞) = D is injective. Thus \underline{D}, defined by \underline{D} = D̃D, is invertible. Let

G = NE_{−}^{−1}  (106)

be a right coprime factorization. As G ∈ RH^{∞} it follows that E_{−} is a stable polynomial matrix. Thus G̃G = Ẽ_{−}^{−1}ÑNE_{−}^{−1} and a spectral factorization of G̃G is obtained from a spectral factorization of ÑN, i.e. ÑN = Ñ_{−}N_{−}. Such a factorization exists by Theorem 2.13 as we assume N(it) is injective for all real t which implies ÑN > 0 on the imaginary axis. Thus

G_{−}(z) = N_{−}(z)E_{−}(z)^{−1}  (107)

and hence,comparing (106) and (107) and as

G̃_{−}(∞)G̃_{−}(∞) = G̃(∞)G̃(∞) = D̃D = \underline{D}

we must have G_{−}(∞) = \underline{D}^{\frac{1}{2}} and G_{−} is realized by [A, B, C_{0}, \underline{D}^{\frac{1}{2}} ], i.e.

G_{−}(z) = \underline{D}^{\frac{1}{2}} + C_{0}(zI − A)^{−1}B.

This implies

G̃_{−}(−z)G_{−}(z) = (\underline{D}^{\frac{1}{2}} − B̃(zI + Ã)^{−1}C̃_{0})(\underline{D}^{\frac{1}{2}} + C_{0}(zI − A)^{−1}B) (108)

A product of transfer functions is realized by the series coupling and so G̃_{−}(−z)G_{−}(z) is realized by

[( \begin{matrix}A & 0 \\ C̃_{0}C_{0} & −Ã \end{matrix} ) , ( \begin{matrix}& B \\ C̃_{0} & \underline{D} & \frac{1}{2} \end{matrix} ) , ( \underline{D} \frac{1}{2} C_{0} −B̃ ) , \underline{D}] .  (109)

But on the other hand the equality G_{∗}G = G_{−∗}G_{−} shows that there is an equivalent realization given by

[( \begin{matrix}A & 0 \\ C̃C & −Ã \end{matrix} ) , ( \begin{matrix}B \\ C̃D \end{matrix} ) , ( D̃C −B̃ ) , D̃D] .  (110)

The two realizations (109) and (110) are isomorphic. The map ( \begin{matrix}I & 0 \\ X & I \end{matrix} )

defines an isomorphism of the two realizations if and only if

( \begin{matrix}I & 0 \\ X & I \end{matrix} ) ( \begin{matrix}A & 0 \\ C̃_{0}C_{0} & −Ã \end{matrix} ) = ( \begin{matrix}A & 0 \\ C̃C & −Ã \end{matrix} ) ( \begin{matrix}I & 0 \\ X & I \end{matrix} )  (111)

and

( D̃C −B̃ ) ( \begin{matrix}I & 0 \\ X & I \end{matrix} ) = ( \underline{D} \frac{1}{2} C_{0} −B̃ ) .  (112)

hold. These two matrix equations reduce to

ÃX + XA + C̃_{0}C_{0} − C̃C = 0  (113)

and

D̃C − B̃X = \underline{D}^{\frac{1}{2}} C_{0}.  (114)

Solving the last equation for C_{0} yields

C_{0} = \underline{D}^{− \frac{1}{2}} (D̃C − B̃X).  (115)

which upon substituting back into equation (113) leads to

\begin{aligned}0 & = & ÃX + XA − C̃C + C̃_{0}C_{0} \\ & = & ÃX + XA − C̃C + (C̃D − XB)\underline{D}^{−1}(D̃C − B̃X). \end{aligned}  (116)

This is the algebraic Riccati equation

X(A−B\underline{D}^{−1}D̃C)+(Ã_{−}C̃D\underline{D}^{−1}B̃)X −C̃C +C̃D\underline{D}^{−1}D̃C +XB\underline{D}^{−1}B̃X = 0.

(117)

This algebraic Riccati equation is associated with the Hamiltonian matrix

( \begin{matrix}A − B\underline{D}^{−1}D̃C & −B\underline{D}^{−1}B̃ \\ −(C̃C − C̃D\underline{D}^{−1}D̃C) & −(Ã − C̃D\underline{D}^{−1}B̃) \end{matrix} )  (118)

From an inspection of the realization (110) it is clear that H is the generator of a realization of (G̃(−z)G(z))^{−1}. But

(G̃(−z)G(z))^{−1} = E_{−}(−z)(Ñ(−z)N(z))^{−1}Ẽ_{−}(z).

So H is symplectically equivalent to S_{ÑN}. The last map clearly has an invariant Lagrangian subspace and therefore so has H. Thus, by Theorem 6.2, the algebraic Riccati equation (117) has a symmetric solution. Moreover there exists a unique stabilizing solution i.e. one for which

A − B\underline{D}^{−1}D̃C − B\underline{D}^{−1}B̃X

is stable.

The next theorem discusses a special case of Theorem 6. in this case the algebraic Riccati equation reduces to a homogeneous one which is equivalent to the Liapunov equation.

Theorem 6.7 Let G be a rational proper transfer function with realization [A, B, C, I]. Assume A is stable and A^{×} = A − BC is antistable. Then G_{∗}G has a spectral factorization G_{−∗}G_{−}, with the spectral factor G_{−} realized by

[A, B, C − B̃X, I]  (119)

and X is the unique stabilizing solution of the homogeneous algebraic Riccati equation

(Ã − C̃B̃)X + X(A − BC) + XBB̃X = 0.  (120)

\underline{Proof:} Let

G(z) = N_{+}(z)E_{−}^{−1}(z)  (121)

then G(z)^{−1} = E_{−}(z)N_{+}(z)^{−1}. By the assumptions of the theorem E_{−} is stable and N_{+} is antistable. A spectral factorization of G_{∗}G is obtained from a spectral factorization Ñ_{−}N_{−} of Ñ_{+}N_{+}. Since Ñ_{+}X_{N_{+}} is just the antistable invariant subspace of S_{Ñ_{+}N_{+}} then the spectral factorization problem reduces to finding the complementary stable invariant subspace. This by the results of section ?? reduces to a Liapunov equation. This we derive directly. From (121) it follows that G_{−} is realized by [A, B, C_{0}, I]. Computing G̃(−z)G(z) = G̃_{−}(−z)_{−}G(z) we have two realizations of G_{∗}G given by

[( \begin{matrix}A & 0 \\ C̃C & −Ã \end{matrix} ) , ( B ; C̃ ) , ( C −B̃ ) , I] .  (122)

and

[( \begin{matrix}A & 0 \\ C̃_{0}C_{0} & −Ã \end{matrix} ) , ( B ; C̃_{0} ) , ( C_{0} −B̃ ) , I] .  (123)

respectively. The map ( \begin{matrix}I & 0 \\ X & I \end{matrix} ) provides an isomorphism of the two real-

izations if and only if the two matrix equations

( \begin{matrix}I & 0 \\ X & I \end{matrix} ) ( \begin{matrix}A & 0 \\ C̃_{0}C_{0} & −Ã \end{matrix} ) = ( \begin{matrix}A & 0 \\ C̃C & −Ã \end{matrix} ) ( \begin{matrix}I & 0 \\ X & I \end{matrix} )  (124)

and

( C −B̃ ) ( \begin{matrix}I & 0 \\ X & I \end{matrix} ) = ( C_{0} −B̃ ) .  (125)

hold. These two matrix equations reduce to

C_{0} = C − B̃X  (126)

and

ÃX + XA + C̃_{0}C_{0} − C̃C = 0  (127)

Substituting (126) into (127) we obtain the homogeneous Riccati equation

(Ã − C̃B̃)X + X(A − BC) + XBB̃X = 0.  (128)

But, as A−BC is antistable, there exists, by Liapunov’s theorem, a positive definite solution Y to the equation

Y (Ã − C̃B̃) + (A − BC)Y = BB̃  (129)

Letting X = Y^{−1} we get a solution to the homogeneous algebraic Riccati equation.

Assume G ∈ RH^{∞}. Certainly if γ > ∥G∥_{∞} then γ^{2}I − G_{∗}G > 0 on iR. Thus a spectral factorization exists. The following result gives a state space formula for the spectral factor of G.

Theorem 6.8 Let G ∈ RH^{∞} have the realization [A, B, C, D] and let γ > ∥G∥_{∞}. Let \underline{D} = (γ^{2}I − D̃D)^{\frac{1}{2}} . Then G_{∗}G has a spectral factorization G_{−∗}G_{−}, with the spectral factor G_{−} realized by

[A, B, \underline{D}^{−1}(D̃C − B̃X), \underline{D}]  (130)

where X is the unique ((stabilizing)) solution of the algebraic Riccati equation associated with the Hamiltonian matrix

( \begin{matrix}A − B\underline{D}^{−2}D̃C & −B\underline{D}^{−2}B̃ \\ −(C̃C − C̃D\underline{D}^{−2}D̃C) & −(Ã − C̃D\underline{D}^{−2}B̃) \end{matrix} )  (131)

\underline{Proof:} Assume G(z) = N(z)E_{−}(z)^{−1} and by our assumption E_{−} is stable. As

γ^{2}I − G̃(−z)G(z) = γ^{2}I − (D − B̃(zI + Ã)^{−1}C̃)(D + C(zI − A)^{−1}B) (132)

it follows that γ^{2}I − G̃(−z)G(z) is realized by

[( \begin{matrix}A & 0 \\ C̃C & −Ã \end{matrix} ) , ( \begin{matrix}B \\ C̃D \end{matrix} ) , ( −D̃C B̃_{−} ) , γ^{2}I − D̃D] .  (133)

On the other hand we have also

\begin{aligned}γ^{2}I − G̃_{∗}G & = & γ^{2}I − (E_{−∗})^{−1}N_{∗}NE_{−}^{−1} \\ & = & (E_{−∗})^{−1}[γ^{2}E_{−∗}E_{−} − N_{∗}N]E_{−}^{−1} \end{aligned}  (134)

and as the middle factor is positive on iR we have a spectral factorization

γ^{2}E_{−∗}E_{−} − N_{∗}N = N_{−∗}N_{−}.  (135)

Hence

γ^{2}I − G̃_{∗}G = [(E_{−∗})^{−1}N_{−∗}][NE_{−}^{−1}] = G_{−∗}G_{−}.  (136)

From the representation G_{−} = N_{−}E_{−}^{−1} it follows that G_{−} has a realization G_{−}(z) = \underline{D} + C_{0}(zI − A)^{−1}B with \underline{D} = (γ^{2}I − D̃D)^{\frac{1}{2}} . So

γ^{2}I−G̃(−z)G(z) = G_{−∗}(z)G_{−}(z) = (\underline{D}−B̃(zI+Ã)^{−1}C̃_{0})(\underline{D}+C_{0}(zI−A)^{−1}B).

(137)

This implies the realization

[( \begin{matrix}A & 0 \\ C̃_{0}C_{0} & −Ã \end{matrix} ) , ( \begin{matrix}& B \\ C̃_{0} & \underline{D} \end{matrix} ) , ( \underline{D}C_{0} −B̃ ) , \underline{D}] .  (138)

The two realizations are isomorphic and we try an isomorphism of the form

( \begin{matrix}I & 0 \\ X & I \end{matrix} ) This map will indeed be an isomorphism provided the following

equalities

( \begin{matrix}I & 0 \\ X & I \end{matrix} ) ( \begin{matrix}A & 0 \\ C̃_{0}C_{0} & −Ã \end{matrix} ) = ( \begin{matrix}A & 0 \\ C̃C & −Ã \end{matrix} ) ( \begin{matrix}I & 0 \\ X & I \end{matrix} )  (139)

and

( D̃C_{0} −B̃ ) ( \begin{matrix}I & 0 \\ X & 0 \end{matrix} ) = ( \underline{D}C_{0} −B̃ ) .  (140)

hold. These two matrix equations imply

ÃX + XA + C̃_{0}C_{0} − C̃C = 0  (141)

and

D̃C − B̃X = \underline{D}C_{0}.  (142)

Solving the last equation for C_{0} yields

C_{0} = \underline{D}^{−1}(D̃C − B̃X).  (143)

which upon substituting back into equation (141) leads to

X(A−B\underline{D}^{−2}D̃C)+(Ã_{−}−C̃D\underline{D}^{−2}B̃)X−C̃C+C̃D\underline{D}^{−2}D̃C+XB\underline{D}^{−2}B̃X = 0,

(144)

i.e. this is the algebraic Riccati equation associated with the Hamiltonian matrix

H = ( \begin{matrix}A − B\underline{D}^{−2}D̃C & −B\underline{D}^{−2}B̃ \\ −(C̃C − C̃D\underline{D}^{−2}D̃C) & −(Ã − C̃D\underline{D}^{−2}B̃) \end{matrix} )  (145)

Now H is symplectically equivalent to the map S_{γ^{2}E_{−∗}E_{−}−N_{∗}N} and it has therefore a Lagrangian invariant subspace. Moreover there is a unique such subspace to which the restriction is stable. As H is the generator of the inverse of the realization of γ^{2}I − G_{∗}G for

H = ( \begin{matrix}A & 0 \\ C̃C & −Ã \end{matrix} ) − ( \begin{matrix}B \\ C̃D \end{matrix} ) \underline{D}^{−2} ( −D̃C B̃ )  (146)

7 INNER-OUTER FACTORIZATIONS

Inner-outer factorizations of analytic functions belonging to vectorial Hardy spaces play an important role both in operator theory as well as its applications in the analysis of infinite dimensional systems and in modern H^{∞} control theory, more specifically in the various forms of the model matching problem.

In this section we study a special case of inner-outer factorizations in the case that the function to be factorred is in RH^{∞}. We will deal here only with the square case and we modify the definitions accordingly.

Thus a function G ∈ RH^{∞} will be called inner if G_{∗}G = I on the imaginary axis, i.e. it is unitary there. A function G ∈ RH^{∞} will be called outer if GH^{2}(C^{n}) is dense in H^{2}(C^{n}). However we will restrict our usage of the term outer to functions G that have an inverse in H^{∞}(C^{n}). This restriction excludes the possibility of zeroes on the imaginary axis.

Under these terms the product of an inner function and an outer function is invertible in the right half plane with the exception of at most a finite number of points.

The next theorem focuses on factorization resuls and state space formulas for the factors.

Let G ∈ RH^{∞} be a rational matrix, which we assume to satisfy G(∞) = I and let G have a realization [A, B, C, I]. Let G = NE_{−}^{−1} be a right coprime factorization. As G ∈ RH^{∞}, detE_{−} has all its zeroes in the open left half plane, i.e. detE_{−} is a stable polynomial. Let N = D_{+}N_{−} be a factorization of N such that detD_{+} has its zeroes in the right half plane whereas detN_{−} in the left half plane. Let G_{i} = D_{+}\overline{D}_{−}^{−1} be the inner factor of G. Then we can write

G_{i} = (D_{+}\overline{D}_{−}^{−1})(\overline{D}_{−}N_{−}E_{−}^{−1}).

Now

\begin{aligned}G_{i} & = & D_{+}\overline{D}_{−}^{−1} \\ & = & (D_{+}N_{−})(N_{−}^{−1}\overline{D}_{−}^{−1}) \\ & = & (D_{+}N_{−})(\overline{D}_{−}N_{−})^{−1} \end{aligned}  (147)

Note that the associated realization corresponding to the last matrix fraction representation of G_{i} is reachable but not necessarily observable. Since G_{i} is a bicausal isomorphism we have

G_{i}^{−1} = (\overline{D}_{−}N_{−})(D_{+}N_{−})^{−1}.  (148)

Comparing (148) with

G^{−1} = (E_{−})(D_{+}N_{−})^{−1}.  (149)

and noting that G^{−1} = [A − BC, B, −C, I], it follows that G_{i}^{−1} = [A − BC, B, C_{0}, I]. Using again the realization of inverse system it follows that [A − B(C + C_{0}), B, −C_{0}, I] realizes G_{i}. Putting F = −(C + C_{0}) the inner function G_{i} is realized by (A + BF, B, C + F, I). Next we proceed to characterize F.

Since G_{i} is an inner function, it satisfies G_{i}^{−1} = G_{i∗}. Now from the realization [A + BF, B, C + F, I] of G_{i} it follows that G_{i∗} is realized by [−(Ã + BF), (C̃ + F), −B̃, I] . On the other hand G_{i}^{−1} is, by the characterization of inverse systems, realized by

[A + BF − B(C + F), B, −(C + F), I] = [A − BC, B, −(C + F), I].

We look for a possible isomorphism X between the two systems, i.e. an invertible map for which the diagram

\begin{aligned}& & & R^{n} \\ & B & ↙ & & ↘ & C̃ + F̃ \\ & & & X \\ & R^{n} & & ⟶ & & R^{n} \\ A − BC & ↓ & & & & ↓ & −(Ã + F̃B̃) \\ & & & X \\ & R^{n} & & ⟶ & & R^{n} \\ & & ↘ & & ↙ \\ & −(C + F) & & R^{n} & & −B̃ \end{aligned}  (150)

is commutative.

The commutativity of this diagram is equivalent to the equations

X(A − BC) + (Ã + F̃B̃)X = 0  (151)

and

− B̃X = −(C + F).  (152)

Thus F = −C + B̃X which substituted back into equation (151) leads to

\begin{matrix}X(A − BC) + (Ã + (XB − C̃)B̃)X \\ = X(A − BC) + (Ã − C̃B̃)X + XBB̃X = 0. \end{matrix}  (153)

The algebraic Riccati equation (153) is associated with the Hamiltonian matrix

H = ( \begin{matrix}A − BC & B̃B \\ 0 & −(Ã − C̃B̃) \end{matrix} )  (154)

However

( \begin{matrix}A − BC & B̃B \\ 0 & −(Ã − C̃B̃) \end{matrix} ) = ( \begin{matrix}A & 0 \\ −C̃C & −Ã \end{matrix} ) − ( \begin{matrix}B \\ −C̃ \end{matrix} ) ( C B̃ )

(155)

But if we compute the series realization of G_{∗}G we obtain

[( \begin{matrix}A & 0 \\ C̃C & −Ã \end{matrix} ) , ( \begin{matrix}B \\ −C̃ \end{matrix} ) , ( C B̃ ) , I] .  (156)

This implies that (G_{∗}G)^{−1} has a realization

[( \begin{matrix}A − BC & B̃B \\ 0 & −(Ã − C̃B̃) \end{matrix} ) , ( B ; C̃ ) , − ( C B̃ ) , I] .  (157)

But on the other hand (G_{∗}G)^{−1} = E_{−}(N_{∗}N)^{−1}E_{−∗}. Thus the algebraic Riccati equation (153) is solvable if and only if N_{∗}N has a spectral factorization. Clearly such a factorization exists if N_{∗}N > 0 on iR. This is guarranteed if we assume the injectivity of N(z) on the imaginary axis.

Theorem 7.1 Let the square rational proper transfer function G ∈ RH^{∞} be injective on iR have a realization [A, B, C, I]. Then G_{∗}G has a spectral factorization G_{−∗}G_{−}, with the spectral factor G_{−} realized by

[A, B, C − B̃X, I]  (158)

and X is the unique stabilizing solution of the homogeneous algebraic Riccati equation

(Ã − C̃B̃)X + X(A − BC) + XBB̃X = 0  (159)

associated with the Hamiltonian matrix

( \begin{matrix}A − BC & BB̃ \\ 0 & −(Ã − C̃B̃) \end{matrix} )  (160)

An inner-outer factorization exists. The outer and inner factors are given by

G_{o}(z) = G_{−}(z)  (161)

and

G_{i}(z) := [A + BF, B, C + F, I]  (162)

respectively, where F = −C + B̃X.

References

[1979] A. C. Antoulas, ”A polynomial matrix approach to F mod G-invariant subspaces”, Doctoral Dissertation, Dept. of Mathematics, ETH Zurich.

[1979] H. Bart, I. Gohberg and M. A. Kaashoek, Minimal Factorization of Matrix and Operator Functions, Birkhauser.

[1979] H. Bart, I. Gohberg, M. A. Kaashoek and P. Van Dooren, ”Factorizations of transfer functions”, .

[1988] T. Chen and B. Francis, ”Spectral and inner-outer factorizations of rational matrices”, to appear.

[1974a] W. A. Coppel, ”Matrix quadratic equations”, Bull. Austr. Math. Soc., 10,377-401.

[1974b] W. A. Coppel, ”Matrices of rational functions”, Bull. Austral. Math. Soc. 11,89-113.

[1980] E. Emre, ”Nonsingular factors of polynomial matrices and (A,B)- invariant subspaces”, SIAM J. Contr. Optimiz. 18,288-296.

[1980] E. Emre and M. L. J. Hautus, ”A polynomial characterization of (A,B)-invariant and reachability subspaces”, SIAM J. Contr. Optimiz., 18, 420-436.

[1988] B. Francis, ”H^{∞} control Theory”, Springer.

[1976] P. A. Fuhrmann, ”Algebraic system theory: An analyst’s point of view”, J. Franklin Inst., 301,521-540.

[1977] P. A. Fuhrmann, ”On strict system equivalence and similarity”, Int. J. Contr. 25,5-10.

[1978] P. A. Fuhrmann, ”Simulation of linear systems and factorization of matrix polynomials”, Int. J. Contr., 28,689-705.

[1979] P. A. Fuhrmann ”Linear feedback via polynomial models”, Int. J. Contr. 30,363-377.

[1981] P. A. Fuhrmann, ”Duality in polynomial models with some applications to geometric control theory,” Trans. Aut. Control, AC-26,284-295.

[1983] P. A. Fuhrmann, ”On symmetric rational transfer functions”, Linear Algebra and Appl., 50,167-250.

[1984] P. A. Fuhrmann, ”On Hamiltonian transfer functions”, Lin. Alg. Appl., 84, 1-93.

[1983] P. A. Fuhrmann, ”A matrix Euclidean algorithm and matrix continued fractions”, Systems and Control Letters, 3, 263-271.

[1984] P. A. Fuhrmann, ”The algebraic Riccati equation - a polynomial approach”, Systems and Control Letters, .

[1979] P. A. Fuhrmann and J. C. Willems, ”Factorization indices at infinity for rational matrix functions”, Integral Equat. and Oper. Theory, 2,287-301.

[1980] P. A. Fuhrmann and J. C. Willems, ”A study of (A,B)-invariant subspaces via polynomial models”, Int. J. Contr. 31,467-494.

[1959] F. R. Gantmacher, The Theory of Matrices , Chelsea, New York.

[1982] I. Gohberg, P. Lancaster and L. Rodman, ”Factorization of selfadjoint matrix polynomials with constant signature”, Lin. and Multilin. Alg., 11, 209-224.

[1978] M. L. J. Hautus and M. Heymann, ”Linear feedback-an algebraic approach”, SIAM J. Control 16,83-105.

[1989] U. Helmke and P. A. Fuhrmann, ”Bezoutians”, to appear, Lin. Alg. Appl..

[1970] V. A. Jacubovich, ”Factorization of symmetric matrix polynomials”, Soviet Math. Dokl., 11, 1261-1264.

[1982] P. P. Khargonekar and E. Emre, ”Further results on polynomial characterization of (F,G)-invariant and reachability subspaces”, IEEE Trans. Aut. Control, 27, 352-366.

[1956] C. C. MacDuffee, The Theory of Matrices , Chelsea, New York.

[1971] K. Martensson, ”On the matrix Riccati equation”, Inform. Sci. 3,17-49.

[1973a] B. P. Molinari, ”The stabilizing solution of the algebraic Ricatti equation”, SIAM J. Cont., 11,262-271.

[1973b] B. P. Molinari, ”Equivalence relations for the algebraic Ricatti equation”, SIAM J. Cont., 11,272-285.

[1966] J. E. Potter, ”Matrix quadratic solutions”, SIAM J. Appl. Math., 14,496-501.

[1984] T. Shamir and P. A. Fuhrmann, ”Minimal factorizations of rational matrix functions in terms of polynomial models”, , Lin. Alg. Appl.

[1983] M. A. Shayman, ”Geometry of the algebraic Ricatti equation, Part I”, SIAM J. Cont., 21,375-394.

[1971] J. C. Willems, ”Least squares stationary optimal control and the algebraic Ricatti equation”, Trans. Automat. Contr., 16,621-634.

[1966] D. C. Youla and P. Tissi, ”N-port inthesis via reactance extractionpart I”, IEEE Inter. Convention Record, 183-205.