ALGEBRAIC METHODS IN SYSTEM THEORY

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

Dedicated to R. E. Kalman, on the occasion of his 60th birthdate.

May 27, 1996

Abstract

The role of algebraic methods in general and module theory in particular as a unifying framework in the theory of linear systems will be surveyed. We will focus on polynomial algebra, realization theory, stability and applications of continued fractions in this area.

1 INTRODUCTION

A Persian rug is valued for its beauty, the originality of its design, the harmony of its colouring and the density of the knots. It turns out that a valuable rug is also very practical and connoisseurs value its usefulness. An area of science is not that different. We are guided by a combination of different criteria. Even if applications are our motivation, it is wise to be guided by aesthetic considerations, it is important to realize precisely the structural properties, and evaluate how any particular part of science is related to the main body. One of the landmarks of a successfull scientific domain is its rich connectivity to other areas. Isolation implies low oxygen supply which rapidly leads to sterility and decay.

It seems necessary to resort to some kind of imagery as for the young newcomers of our field, exposed very early to a bombardment of highly technical and sophisticated mathematics, it is hard to realize how the field of system and control reached this point.

Clearly system theory is the result of a joint effort of many scientists, going back at least to J. C. Maxwell in the preceeding century. However the time things started to move from the vague to the precise, from the computational to the conceptual i.e. from problem solving to the creation of a theory, is clearly the late fifties and early sixties and in this process the dominant figure was no doubt R. E. Kalman. His contributions were, from the technical point, superceeded quickly by people better trained in their respective fields. However in no case was the technical contribution the important part. The importance related to the tremendous insight into the heart of the problem. Thus with optimal control and the fundamental new point of view in filtering. The same is true with the conceptual foundation of the theory of finite dimensional linear systems. Thus the focusing on controllability as a fundamental new concept, its relation to realization theory, the isomorphism theorem and the grand unification of the external and internal viewpoints in system theory through the use of module theory.

I have already in the past, in Fuhrmann [1976], indicated my debt to Kalman’s work in system theory. I have profited mostly from the process of algebraization that Kalman brought to the field and it is in this area that I would like to confine my contribution to this volume.

Algebra historically was almost synonimous with the theory of equations.

Linear algebra was concerned mostly with systems of linear equations and the formal manipulation of matrices and determinants. In the twentieth century algebra underwent a major revolution, and the essence of the revolution was the change of focus from the special to the general, the change of emphasis from the problem solving to the structure and the conceptual understanding. The influence of Hilbert and E. Noether in this process was all pervasive. The change was announced and widely recognized with the appearance of van der Waerden [1931] treatise. Not by accident did module theory play an important role in this development. In fact here, I believe for the first time, was a treatment of the structure of linear transformations based on the naturally induced module structure.

The module theoretic approach to linear algebra, it has no competition in terms of power and elegance, did still not find general acceptance in undergraduate mathematical, not to say engineering, curricula. Thus the importation by Kalman of module theory as a working tool in system theory was nothing short of revolutionary. The naturality of the concept to this setup account no doubt for the great success it had in providing a unifying point of view.

In a short contribution it is difficult to do justice to the many different ways that module theory can be applied to problems of systems and control. So, rather than give a systematic account, at this stage even a reasonable size book may not be up to the task, I will present a series of vignettes which, if they arouse the reader’s interest, can be used as starting points for further reading and study. Each of the topics chosen is related in one way or another to previous work by Kalman.

2 MODULAR ARITHMETIC

In Kalman [1969] the use of algebra as a computational tool, extremely well suited for computers, is stressed. This point really raises the question of how abstract can one make a theory while retaining the ability for easy computability. Clearly this is related to what are the most convenient representations of abstract objects. The situation is analogous to the case of linear transformations in finite dimensional vector spaces and their representations as matrices. However, even in this case, this well established representation is by no means the only one, and the point I will try to make, not even the most convenient, certainly not the most illuminating from the structural point of view. The importance of polynomial algebra in this connection has been realized early by Kalman, not only as a key to structure via the use of polynomial modules, but also as a convenient computational tool, namely through the use of modular arithmetic. This of course was possible if one restricted to the scalar case, i.e. to single input/single output systems. Simultaneously the systematic use of polynomial matrices in system theory was begun by Rosenbrock [1970]. While this turned out to be an extremely powerful tool, it paid less attention to the fundamental conceptual underpinnings of the subject.

I was in a lucky position, see Fuhrmann [1976], to be able to realize that Kalman’s emphasis of the abstract theory of modules and Rosenbrock’s use of coprime factorizations and polynomial matrices, the generally available state space methods, as well as the theory of infinite dimensional systems beginning its development at that time, were all different facets that could be unified in one approach. Specifically one goes from the abstract theory of modules to certain polynomial representations. This leads to a modular arithmetic on polynomial vectors and matrices. Finally looking at the new objects, namely polynomial models, any choice of linear basis leads to matrix or state space representations. From this point of view the use of polynomial models is a balancing act between the abstract and the concrete.

From an external point of view regarding systems, an input output map was defined by Kalman as a module homomorphism, over the ring of polynomials, between appropriately defined spaces of input and output functions. Of course this implies a linear context and the module property implies time invariance. The causality property can be introduced by requiring the invariance of an appropriate submodule under the input/output map. Alternatively one can implicitly introduce causality by considering the resrticted input/output map

f : Ω ⟶ Γ.  (1)

where Ω is the space of past input functions and Γ the space of future output functions. The assumptions on the various objects appearing has a profound influence on the development of the theory. Kalman’s choice was Ω = F^{m}[z] and Γ = z^{−1}F^{p}[[z^{−1}]]. We shall study these and related objects in some detail and return to Kalman’s abstract realization in section 5 It is convenient to take a slightly more general setting.

Let F denote an arbitrary field. We will use the following notation. F [z] will denote the ring of polynomials over F, F (z) the field of rational functions and F [[z^{−1}]] the ring of formal power series in z^{−1} with coefficients in F, i.e. f ∈ F [[z^{−1}]] then f(z) = ∑_{j=0}^{n} f_{j}z^{j}. By z^{−1}F [[z^{−1}]] we denote the subspace of F [[z^{−1}]] consisting of all power series with vanishing constant term and by F ((z^{−1})) we denote the field of truncated Laurent series namely of series of the form

h(z) = ∑_{j=−∞}^{n(h)} h_{j}z^{j} with n(h) ∈ Z.

An element h ∈ F ((z^{−1})) is called rational if there exists a nonzero polynomial q such that qh ∈ F [z]. Thus every rational function has a representation of the form h = p/q with p and q polynomials. An invertible element of F [[z^{−1}]] will be called a bicausal isomorphism.

In much the same way we can introduce the corresponding spaces, R((z^{−1})), R[z] and R[[z^{−1}]] where R is an arbitrary ring. Even more generally, for a left R module M, we define M((z^{−1})), M[z] and M[[z^{−1}]] as before. While addition is defined in M((z^{−1})) as usual multiplication is not. But the multiplication of elements of R((z^{−1})) and M((z^{−1})) is defined. M((z^{−1})) becomes a module over various rings including R, R[z], R((z^{−1})) and R[[z^{−1}]]. We will consider here only the case of M = F^{m}. Various module structures can be introduced. In particular F^{m}[z] is an R[z] submodule of F^{m}((z^{−1})), and as F [z] modules we have the following short exact sequence of module homomorphism

0 ⟶ F^{m}[z] ⟶^{j} F^{m}((z^{−1})) ⟶^{π} F^{m}((z^{−1}))/F^{m}[z] ⟶ 0

with j the embedding of F^{m}[z] into F^{m}((z^{−1})) and π the canonical projection onto the quotient module.

Elements of F^{m}((z^{−1}))/F^{m}[z] are equivalence classes and two elements are in the same equivalence class if and only if they differ in their polynomial terms only. Thus a natural choice of element in each equivalence class is the one element whose polynomial terms are all zero. This leads to the identification of F^{m}((z^{−1}))/F^{m}[z] with z^{−1}F^{m}[[z^{−1}]]. With this identification we denote by π_{−} the canonical projection, i.e.

π_{−} ∑_{j=−∞}^{n} f_{j}z^{j} = ∑_{j=0}^{n} f_{j}z^{j}.

Since Kerπ_{−} = F^{m}[z] and Imπ_{−} = z^{−1}F^{m}[[z^{−1}]] we have a direct sum decomposition, over R,

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

We will denote by π_{+} the complementary projection on F^{m}[z], i.e.

π_{+} = 1 − π_{−}

or equivalently

π_{+} ∑_{j} f_{j}z^{j} = ∑_{j≥0} f_{j}z^{j}.

At this point we will introduce a special operator, namely the shift operator S defined by

(Sf)(z) = zf(z) for f ∈ V ((z^{−1})).

Clearly S is a linear map that is invertible and S^{−1}f = z^{−1}f. The name derives from the representation of the map in terms of the sequence of coefficients. Indeed if f(z) = ∑ f_{j}z^{j} and we make the correspondence

f(z) ↔ (…, f_{2}, f_{1}, \underline{f}_{0}, f_{−1}, …)

where the underlined term is the coefficient of z^{0} then

Sf(z) ↔ (…, f_{1}, f_{0}, \underline{f}_{−1}, f_{−2}, …).

Since F^{m}[z] is a submodule, relative to F^{m×m}[z], then in particular SF^{m}[z] ⊂ F^{m}[z], i.e. it is an S−invariant subspace. Therefore we can define the restriction of S to F^{m}[z] and denote it by S_{+}, that is

S_{+} = S|F^{m}[z].

Similarly S induces an F −linear map in z^{−1}F^{m}[[z^{−1}]] defined by

S_{−}h = π_{−}zh for h ∈ z^{−1}F^{m}[[z^{−1}]].

We note that S_{+} is injective but not surjective whereas S_{−} is surjective but not injective.

The map S_{−} has many eigenfunctions. In fact, each α in F is an eigenvalue of S_{−} , with the eigenfunctions given by

v(z) = (z − α)^{−1}ξ ξ ∈ F^{m}.

In the same way one can show the existence of generalized eigenfunctions of arbitrary order. Contrary to this richness of eigenfunctions, the shift S_{+} does not have any eigenfunctions. The previous theorem indicates that the spectral structure of the shift S_{−}, with finite multiplicity, is rich enough to model all finite dimensional linear transformations, up to similarity transformations. This is indeed the case as the next theorem shows.

Theorem 2.1 (Rota) Let A be a linear transformation in finite dimensional vector space F^{m} over F, which we take to be F^{m}. Then A is isomorphic to S_{−} restricted to a finite dimensional S_{−}-invariant subspace of z^{−1}F^{m}[[z^{−1}]].

\underline{Proof:} Define the set L by L = \{(zI − A)^{−1}ξ|ξ ∈ F^{m}\}. Since

(zI − A)^{−1}ξ = ∑_{j=1}^{∞} A^{j}ξz^{−(j+1)}

L is a subspace of z^{−1}F^{m}[[z^{−1}]]. Moreover since

\begin{matrix}S_{−}(zI − A)^{−1}ξ = π_{−}z(zI − A)^{−1}ξ \\ = π_{−}(zI − A + A)(zI − A)^{−1}ξ = (zI − A)^{−1}Aξ \end{matrix}

is in L it is clearly S_{−}-invariant. Finally if ϕ : F^{m} ⟶ L is defined by ϕξ = (zI − A)^{−1}ξ then ϕ is injective and ϕA = S_{−}ϕ , so A is isomorphic to S_{−}|L with L = Imϕ.

Considerations of duality suggest that rather than use the restriction of S_{−} to submodules as models for finite dimensional linear transformations, we can use compressions of S_{+} to quotient modules of F^{m}[z]. A compression of S_{+} is the map induced by S_{+} in a quotient module. This we proceed to show directly. The result is dual to Rota’s theorem.

Theorem 2.2 Let A be a linear transformation in the finite dimensional vector space F^{m} over the field F. Then A is isomorphic to the map induced by S_{+} in a quotient module of F^{m}[z].

\underline{Proof:} Define a map Φ : F^{m}[z] ⟶ F^{m} by

Φ ∑_{j=0}^{n} v_{j}z^{j} = ∑_{j=0}^{n} A^{j}v_{j}.

Then clearly Φ is surjective and satisfies

ΦS_{+} = AΦ.

If an F [z]−module structure is defined on F^{m} by

p · v = p(A)v for p ∈ F [z]

then Φ is actually a surjective module homomorphism. Thus we have the F [z]−module isomorphism

F^{m} ≃ F^{m}[z]/KerΦ

and we are done.

It is of interest to obtain an explicit description of KerΦ. We will show that KerΦ = (zI − A)F^{m}[z]. To this end let v ∈ (zI − A)F^{m}[z] then

v(z) = (zI − A)w(z) = (zI − A) ∑_{j=0}^{n} w_{j}z^{j} = ∑_{j=0}^{n} w_{j}z^{j+1} − ∑_{j} Aw_{j}z^{j}.

So

Φ(v) = ∑_{j=0}^{n} A^{j+1}w_{j} − ∑_{j=0}^{n} A^{j}Aw_{j} = 0.

Conversely let v ∈ KerΦ, then if v(z) = ∑_{j=0}^{n} v_{j}z^{j}, we have ∑_{j=0}^{n} A^{j}v_{j} = 0. This implies that

v(z) = ∑_{j=0}^{n} v_{j}z^{j} − ∑_{j=0}^{n} A^{j}v_{j} = ∑_{j=0}^{n}(z^{j}I − A^{j})v_{j}.

But

(z^{j}I − A^{j}) = (zI − A)(z^{j−1}I + z^{j−2}A + ⋯ + A^{j−1})

and so v(z) = (zI − A)w(z).

The moral of this is that in the quotient module F^{m}[z]/KerΦ the submodule KerΦ has a nice representation namely KerΦ = (zI − A)F^{m}[z] and moreover the compression of the shift to the corresponding quotient module is isomorphic to A. This suggests that we study the lattice of submodules of F^{m}[z]. This we do in the next section.

3 COPRIMENESS

In this section we focus on coprimeness. This is a classical topic, going back to Greek mathematics that culminated in the Euclidean algorithm. In the fabric of system theory the presence of the notion of coprimeness and the Euclidean algorithm, and the related Bezout equation, are all pervasive. They relate to the geometry of invariant subspaces, spectral problems and inversion of operators, stability criteria, canonical forms, controllability criteria, continued fraction expansions, recursive algorithms, and this is a partial list.

We will introduce coprimeness on the level of polynomial matrices, relate these to geometric ideas as well as to spectral representations. Finallly we will apply it to the study of inversion of the module homomorphisms characterized in Theorem 3.6.

All these ideas have their counterpart in functional analysis and operator theory. In its most far reaching extension the idea of coprimeness is present in the celebrated corona theorem, proved by Carleson [1962]. For the application of this to spectral theory see Fuhrmann [1968], and to infinite dimensional realization theory we refer to Fuhrmann [1981]. Recently a very nice application to the robust stabilization problem has been provided by Georgiou and Smith [1990]. For an effort to present a coherent connection between polynomial coprimeness and H^{∞} coprimeness see Fuhrmann [1989].

We begin now with the arithmetization of the lattice operations in the set of submodules of F^{m}[z]. First we recall some definitions. Given elements D_{i} in a ring R then D is a common left divisor , or common left factor if there exist E_{i} in R such that D_{i} = DE_{i}. E is a common right multiple of the D_{i} if there exist E_{i} such that D_{i}E_{i} = E. D is a greatest common right divisor , if it is a common right divisor and is a right multiple of any other right divisor. Elements D_{i} are left coprime if their g.c.l.d. is a unit. The symmetric concepts are similarly defined.

The set of submodules of F^{m}[z] is partially ordered by inclusion. Inclusion is related to factorization of the representing polynomial matrices.

Theorem 3.1 Let M = DF^{m}[z] and N = EF^{m}[z] then M ⊂ N if and only if D = EG for some G in F^{p×m}[z].

Two natural operations on submodules are intersection and sums and it is of interest to relate these operations to the representatives of the corresponding submodules.

Theorem 3.2 Given submodules M_{i} = D_{i}F^{m}[z], i = 1, …, s let ∩_{i=1}^{s}M_{i} = DF^{m}[z] and ∑_{i=1}^{s} M_{i} = EF^{m}[z]. Then D is a l.c.r.m. of the D_{i} and E is a g.c.l.d. of the D_{i}.

Corollary 3.1 (i) Given D_{i}, i = 1, …, s, in F^{p×m}[z] then the D_{i} have a g.c.l.d. D which can be expressed as

D = ∑_{i=1}^{s} D_{i}E_{i}.

(ii) Given D_{i}, i = 1, …, s, in F^{p×m}[z] then the D_{i} have a g.c.r.d. D which can be expressed as

D = ∑_{i=1}^{s} E_{i}D_{i}.

Corollary 3.2 (i) Elements D_{i}, i = 1, …, s, in F^{p×m}[z] are left coprime if and only if there exist E_{i} such that

∑_{i=1}^{s} D_{i}E_{i} = I.

(ii) Elements D_{i}, i = 1, …, s, in F^{p×m}[z] are right coprime if and only if there exist E_{i} such that

∑_{i=1}^{s} E_{i}D_{i} = I.

Set operations on invariant subspaces of X_{D} are reflected in the factorizations. This is summed up in the following.

Theorem 3.3 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}.

The previous theorem leads immediately to the problem of decomposing a polynomial model into a direct sum of submodules.

Theorem 3.4 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.

We saw, in Theorem 2.2, how every linear transformation A is isomorphic to the compression of the shift S_{+} to a quotient module F^{m}[z]/DF^{m}[z] with D(z) = zI − A. The computational inconvenience of working with equivalence classes can be removed if we can find a good way of choosing a representative in each equivalence class of F^{m}[z]/DF^{m}[z]. Let us analyze the scalar case. We consider the quotient module F [z]/dF [z] with d a nonzero polynomial. Now given any f in F [z] we can write f = ad + r and this representation is unique if degr < degd. To isolate the remainder r we proceed by dividing the previous equality by d, i.e. fd^{−1} = a + rd^{−1}. Now a is in F [z] whereas rd^{−1}, since degr < degd, is in z^{−1}F [[z^{−1}]] so applying the projection π_{−} we have π_{−}d^{−1}f = rd^{−1} and r = dπ_{−}d^{−1}f. The advantage of this circuitious route to the remainder r is in the ease with which it generalizes to the multivariable case.

Definition 3.1 For a nonsingular polynomial matrix D ∈ F^{m×m}[z] we define the map π_{D} by

π_{D}f = Dπ_{−}D^{−1}f, for f ∈ F^{m}[z].

Theorem 3.5 Let D be a nonsingular polynomial matrix in F^{m×m}[z]. Then π_{D} is a projection in F^{m}[z] and Kerπ_{D} = DF^{m}[z].

We introduce now an F [z]−module structure in X_{D} = Imπ_{D} by letting

p · f = π_{D}(pf)

for all p in F [z] and all f in F^{m}[z].

With the previously defined module structure X_{D} is isomorphic to F^{m}[z]/DF^{m}[z].

In X_{D} we will focus on a special map S_{D} 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}.

The following theorem is a characterization of F[z]-module homomorphisms between polynomial models. This theorem, proved in Fuhrmann [1976], is the algebraic version of the celebrated commutant lifting theorem, due to Sarason [1967], Sz.-Nagy and Foias [1970]. In this connection Nikolskii

[1985] is a convenient reference.

Theorem 3.6 Let D ∈ F^{m×m}[z] and D_{1} ∈ F^{m_{1}×m_{1}}[z] be nonsingular polynomial matrices. Then Z : X_{D} ⟶ X_{D_{1}} is an F [z]−homomorphism, i.e. ZS_{D} = S_{D_{1}}Z, if and only if there exist M and N in F^{m_{1}×m}[z] such that

ND = D_{1}M  (2)

and

Zf = π_{D_{1}}Nf.  (3)

We outline next with the invertibility properties of these homomorphisms. We do this through the characterization of the kernel and image of the map Z defined by equation (3) in terms of the polynomial data (2).

Theorem 3.7 Let Z : X_{D} ⟶ X_{D_{1}} be the module homomorphism defined in Theorem 3.6. Then

(i) KerZ = EX_{G}, where D = EG and G is a g.c.r.d. of D and N.

(ii) ImZ = E_{1}X_{G_{1}}, where D_{1} = E_{1}G_{1} and E_{1} is a g.c.l.d. of D_{1} and M.

(iii) Z is surjective if and only if N and D_{1} are left coprime.

(iv) Z is injective if and only if D and M are right coprime.

4 HANKEL OPERATORS AND MODULE HOMOMORPHISMS

Given a p × m matrix function G ∈ F^{p×m}((z^{−1})) it induces a Hankel map H_{G} : F^{m}[z] ⟶ z^{−1}F^{p}[[z^{−1}]] defined by

H_{G}f = π_{−}Gf, f ∈ F^{m}[z].  (4)

It is easy to check that a Hankel operator satisfies the Hankel functional equation, namely

S_{−}H = HS_{+}.  (5)

Thus solutions of this equation are just module homomorphisms from F^{m} to z^{−1}F^{p}[[z^{−1}]].

The following is characterization of all solutions of the Hankel functional equation, as well as a characterization of rationality. The theorem can be viewed as an algebraic version of Nehari’s theorem, see Nehari [1957] and Nikolskii [1985]. The characterization of finite rank Hankel operators is due to Kronecker [1890].

Theorem 4.1 There exists a map H : F^{m}[z] ⟶ z^{−1}F^{p}[[z^{−1}]] that satisfies (5) if and only if there exists a G ∈ F^{p×m}((z^{−1})) such that H = H_{G}. G is rational if and only ImH_{G} is finite dimensional.

It is natural to inquire whether the two theorems, Theorem 3.6 and Theorem 4.1, each a characterization of module homomorphisms, are related in some way. This turns out to be the case, and in fact this is a basis for a very general and powerful technique in operator theory. For more on this one can consult Nikolskii [1985]. Notice the naturality of the introduction of matrix fraction representations of rational matrices.

Theorem 4.2 Theorems 4.1 and 3.6 are equivalent.

Proof: Assume Theorem 4.1. Let ZS_{D} = S_{D_{1}}. Define a map H : F^{m}[z] ⟶ z^{−1}F^{p}[[z^{−1}]] by

H = D_{1}^{−1}Zπ_{D}.  (6)

We will show that H is a Hankel map.

\begin{aligned}HS_{+}f & = & D_{1}^{−1}Zπ_{D}zf = D_{1}^{−1}Zπ_{D}zπ_{D}f \\ & = & D_{1}^{−1}ZS_{D}π_{D}f = D_{1}^{−1}S_{D_{1}}Zπ_{D}f = D_{1}^{−1}D_{1}π_{−}D_{1}^{−1}zZπ_{D}f \\ & = & π_{−}zD_{1}^{−1}Zπ_{D}f = S_{−}Hf \end{aligned} (7)

So H satisfies the Hankel functional equation (5). By Theorem 4.1 there exists a G such that H = H_{G}. So

π_{−}Gf = D_{1}^{−1}Zπ_{D}f,  (8)

or

D_{1}π_{−}Gf = Zπ_{D}f.

Now, from equation (8) it follows that ImH_{G} ⊂ X^{D_{1}}. Hence

π_{−}D_{1}π_{−}Gf = π_{−}D_{1}Gf = 0.

Since this is true for all vector polynomials f we get D_{1}G = N_{1}, or

G = D_{1}^{−1}N_{1}.  (9)

So, for f ∈ X_{D},

Zf = D_{1}π_{−}D_{1}^{−1}N_{1}f = π_{D_{1}}N_{1}f.

Now, from the definition of H, it follows that KerH ⊃ DF^{m}[z]. So, for any vector polynomial f,

π_{−}GDf = 0,

i.e. N = GD is a polynomial matrix. So

G = ND^{−1}  (10)

and we have

D_{1}^{−1}N_{1} = ND^{−1}.  (11)

Notice that no coprimeness assumptions have been made. Also we note that equation (11) is equivalent to the intertwining relation (2).

Conversely, assume Theorem 3.6 holds. Let H be a Hankel operator and assume ImH is finite dimensional. So obviously H : F^{m}[z] ⟶ ImH is surjective.

Define Z : X_{D} ⟶ X_{D_{1}} by Z = D_{1}H. We claim ZS_{D} = S_{D_{1}}Z. Indeed, for f ∈ X_{D},

\begin{aligned}ZS_{D}f & = & D_{1}HDπ_{−}D^{−1}zf = D_{1}Hzf = D_{1}HS_{+}f \\ & = & D_{1}S_{−}Hf = D_{1}π_{−}zHf = D_{1}π_{−}D_{1}^{−1}zD_{1}Hf = S_{D_{1}}Zf \end{aligned} (12)

By Theorem 3.6, there exist polynomial matrices N, N_{1} such that N_{1}D = D_{1}N and Zf = π_{D_{1}}Nf. So

Hf = D_{1}π_{D_{1}}N_{1}f = π_{−}D_{1}^{−1}N_{1}f = π_{−}Gf.

5 REALIZATION THEORY

We saw, in the proof of Theorem 3.6, the natural way in which matrix fractions arise. We use now these matrix fractions to write down an extremely simple realization procedure. While this is a basis free approach, it goes without saying that special choices of basis lead to special matrix realizations, and in turn to a variety of canonical forms. This is a very general method which we do not explore in full in this paper. However we will give some examples in connection with continued fractions. We begin by recalling some concepts.

Definition 5.1 A discrete time, constant linear system Σ is a triple of F −linear spaces X, U and Y and a triple of maps (A, B, C) with A ∈ L(X, X), B ∈ L(U, X) and C ∈ L(X, Y). The triple of maps represents the system of equations

\begin{matrix}x_{n+1} = Ax_{n} + Bu_{n} \\ y_{n} = Cx_{n} + Du_{n}. \end{matrix}  (13)

The dimension of the system, dimΣ, is defined by

dimΣ = dimX.

The spaces X,U and Y are referred to as the state space , input space and output space respectively.

Here the summation is well defined as the sequence of inputs u_{k} is finitely nonzero for negative indices and hence there are only a finite number of nonzero terms in the summation. The input/output relations induced by the system can therefore be written in the form of a discrete convolution product

y_{n} = ∑_{j=0}^{∞} CA^{j}Bu_{n−j−1}.

We note that the input/output relation depends on the triple (A, B, C) only through the maps CA^{j}B, which are called the Markov parameters of the system.

Realization theory is an inverse problem,. Starting from an i/o map f it is required to construct a state space system Σ = (A, B, C) for which f = f_{Σ}.

In that case we say Σ = (A, B, C) is a realization of f. Let us consider the implications of this definition. In order to do this we recall now the important notions of reachability and observability. These notions were introduced and studied by Kalman. For a full historical discussion we refer to Kalman [1969] and Kalman, Falb and Arbib [1969].

Definition 5.2 Given the system Σ = (A, B, C) with the state space X. A state x ∈ X is called reachable if there exists a sequence of inputs driving the system from the zero state to x. A state x ∈ X is called unobservable if in the absence of inputs all outputs of the system with x as the initial state are zero. We say the system Σ is reachable if every state is reachable and observable if the only unobservable state is the zero state.

We state without proof the following simple criteria for reachability and observability.

Theorem 5.1 The system Σ = (A, B, C) is reachable if and only if

∩_{i=0}^{∞} KerB̃Ã^{i} = {0}  (14)

and observable if and only if

∩_{i=0}^{∞} KerCA^{i} = {0}  (15)

If dimX = n then (14) and (15) can be replaced by

∩_{i=0}^{n−1} KerB̃Ã^{i} = {0}  (16)

and

∩_{i=0}^{n−1} KerCA^{i} = {0}  (17)

respectively.

Conditions (16) and (17) are equivalent to the Kalman reachability and observability rank conditions. Define now the reachability map R : F^{m}[z] ⟶ X and observability map O : X ⟶ z^{−1}F^{p}[[z^{−1}]] of the system Σ = (A, B, C) by

R ∑_{i=0}^{n} u_{i}z^{i} = ∑_{i=0}^{n} A^{i}Bu_{i}

and

Ox = ∑_{i=0}^{∞} \frac{CA^{i}x}{z^{i+1}}.

The next lemma summarizes the basic properties of the reachability and observability maps.

Lemma 5.1 With the module structure induced in X by A, the reachability map R and observability map O of the system Σ are F [z]−module homomorphisms. R is surjective if and only if Σ is reachable and O is injective if and only if Σ is observable.

A system is called canonical or minimal if it is both reachable and observable. In the same way we say that Σ = (A, B, C) is a canonical realization of an input/output map f if it is a reachable and observable realization.

We observe now the following.

Theorem 5.2 Given a restricted i/o map f and a realization Σ = (A, B, C). Then f admits a factorization

f = OR

where R and O are the reachability and observability maps of Σ. In other words the diagram

f
U[z] ⟶ z^{−1}Y [[^{−1}]]
R ↘  ↙ O
X

is commutative.

We note that, with the transfer function G of the system defined by

G(z) = C(zI − A)^{−1}B

the restricted input/output map f is equal to the Hankel map H_{G}.

Now if X is taken to be an F [z]−module with the module structure induced by A. This leads us, following Kalman, to the following definition.

Definition 5.3 Given an F [z]−module homomorphism f : U[z] ⟶ z^{−1}Y [[z^{−1}]]. Then an abstract realization of f is a factorization

f = hg

with g : U[z] ⟶ X and h : X ⟶ z^{−1}Y [[z^{−1}]] two F [z]−homomorphisms. The realization is called reachable if g is surjective and observable if h is injective. It is called canonical if it is both reachable and observable. The realization is finite dimensional if X is finite dimensional as an F −linear space.

Theorem 5.3 Given a restricted i/o map f then canonical realizations of f always exist.

\underline{Proof:} We can always factor f canonically. One way to do it is to factor f as

f
F^{m}[z] ⟶ z^{−1}F^{p}[[^{−1}]]
g ↘  ↗ h
F^{m}[z]/Kerf

where g is the canonical projection and h the homomorphism induced by f on the factor module. An alternative canonical factorization is

f
F^{m}[z] ⟶ z^{−1}F^{p}[[^{−1}]]
g ↘  ↗ h
Imf

where now g differs from f only in the range module and h is the canonical injection of Imf into z^{−1}F^{p}[[z^{−1}]].

Our representation theorems, using the polynomial and rational models X_{D} and X^{D} enable us to pass from the abstract realization of an i/o map f to a more concrete one. For this we use the matrix fraction representation of rational matrices and their relation to the kernel and image of the induced Hankel map.

Let G = ND^{−1} be a matrix fraction representation of a rational G, no coprimeness assumptions being made. Define the shift realization to be the realization (A, B, C) in the state space X_{D} defined by

\{ \begin{matrix}A = S_{D} \\ Bξ = π_{D}ξ & for ξ ∈ F^{m}, \\ Cf = (ND^{−1}f)_{−1} & for f ∈ X_{D} \end{matrix}  (18)

In order to show that this is a realization of G we compute

CA^{i−1}Bξ = (ND^{−1}S_{D}^{i−1}π_{D}ξ)_{−1} = (ND^{−1}Dπ_{−}D^{−1}z^{i−1}π_{D}ξ)_{−1}
= (Nπ_{−}D^{−1}z^{i−1}ξ)_{−1} = (ND^{−1}z^{i−1}ξ)_{−1}
= (z^{i−1}Gξ)_{−1} = (π_{−}z^{i−1}Gξ)_{−1}
= G_{i}ξ,

i.e. we have a realization. We compute next the reachability and observability maps of this realization.

\begin{aligned}R ∑_{i=0}^{n} u_{i}z^{i} & = & ∑_{i=0}^{n} S_{D}^{i}π_{D}u_{i} = ∑_{i=0}^{n} π_{D}z^{i}π_{D}u_{i} \\ & = & ∑_{i=0}^{n} π_{D}z^{i}u_{i} = π_{D} ∑_{i=0}^{n} z^{i}u_{i} \end{aligned}

i.e. R : F^{m}[z] ⟶ X_{D} is given by

Ru = π_{D}u for all u ∈ F^{m}[z].

Obviously, since X_{D} = Imπ_{D}, R is surjective and hence the previous realization is reachable. For the observability map O we have

Of = ∑_{i=1}^{∞}(ND^{−1}π_{D}z^{i−1}f)_{−1}z^{−i} = ∑_{i=1}^{∞}(ND^{−1}Dπ_{−}D^{−1}z^{i−1}f)_{−1}z^{−i}
= ∑_{i=1}^{∞}(ND^{−1}z^{i−1}f)_{−1}z^{−i} = ∑_{i=1}^{∞}(π_{−}z^{i−1}Gf)_{−1}z^{−i} = π_{−}Gf
= H_{G}f.

We can summarize this analysis in the following Theorem 5.4 The realization associated with the matrix fraction G = ND^{−1} is reachable. It is observable if and only if N and D are right coprime.

Next we construct an shift realization associated with the matrix fraction G = D_{1}^{−1}N_{1}. We choose X_{D} as the state space and define [A_{1}, B_{1}, C_{1}] through

\{ \begin{matrix}A_{1} = S_{D} \\ B_{1}ξ = π_{D}N_{1}ξ = N_{1}ξ & for ξ ∈ F^{m}, \\ C_{1}f = (D_{1}^{−1}f)_{−1} & for f ∈ X_{D_{1}}. \end{matrix}  (19)

Theorem 5.5 The realization associated with the matrix fraction G = D_{1}^{−1}N_{1} is observable. It is reachable if and only if N_{1} and D_{1} are left coprime.

We can rewrite this realization using the rational model rather than the polynomial one. Thus the state space is chosen as X^{D} and (A, B, C, D) is defined through

\{ \begin{matrix}A = S^{D} \\ Bξ = D_{1}^{−1}N_{1}ξ & for ξ ∈ R, \\ Cf = (f)_{−1} = (zf)(∞) & for f ∈ X^{D} \\ D = G(∞). \end{matrix}  (20)

It is this realization, in its scalar version, we use as a basis for obtaining a balanced realization.

The two realization procedures outlined above can be combined into a single one by looking at more general representations of rational transfer functions. Thus we will assume the transfer function of a system is given by

G = V T^{−1}U + W.  (21)

Our approach to the analysis of these systems is to associate with each representation of the form (21) 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 : F^{m} ⟶ X_{T}, and C : X_{T} ⟶ F^{p} by

\begin{matrix}A = S_{T} \\ Bξ = π_{T} Uξ, \\ Cf = (V T^{−1}f)_{−1}. \end{matrix}  (22)

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

Theorem 5.6 The system given by (22) is a realization of π_{−}G where 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.

In particular this approach allows us to study realizations that are neither reachable nor observable. For an isomorphism analysis of such realizations and its relation to system equivalence see Fuhrmann [1977]. This more general realization procedure is also more amenable to the study of realizations reflecting some symmetry in the input/output behaviour. For this we refer to Fuhrmann [1983,1984].

6 STABILITY THEORY

Stability theory is, in the context of linear, finite dimensional time invariant systems, concerned with root location of polynomials. The origin of this problem goes back at least to the middle of the previous century, with the work of Jacobi and Borchardt. This work utilized the theory of quadratic forms. This powerful tool reached its perfection at the hands of a master, C. Hermite [1856]. In a completely different direction we have the pioneering work of Liapunov [1893]. This reduces in our case to the analysis of the celebrated Liapunov equation.

Now these two widely differing approaches to the study of stability are manifestastions of a constant phenomena in system theory. The main characterizing object of a time invariant system can be taken to be, in view of realization theory, its transfer function. The transfer function of a finite dimensional system is rational, and a rational function can be viewed in two very different ways. On the one hand we can see it as an algebraic object and apply to its study algebraic methods. On the other hand we can view it as a complex valued function and study it in analytic terms. This Janus like character of system theory accounts in no small part for the richness of the field and the depth of some of the results. In multifaceted situations like this, especially in a dynamic field of research as the theory of dynamical systems, it becomes increasingly difficult to maintain a global encompassing point of view. But whenever such a view can be obtained it is highly profitable and provides some extra insight to the theory.

It is in this context that one can view Kalman’s [1969] contribution to the area of stability.

In this work Kalman set out to give a unified treatment of the classical stability criteria of Hermite, Hurwitz, Schur-Cohn and Liapunov. In the process the algebraic theory of quadratic forms and the Liapunov equation are unified. While this is not the first result relating Liapunov’s method to the method of quadratic forms, apparently the first demonstration of this is due to Parks [1962], it is a highly ingenious one. The approach is algebraic and uses modular arithmetic in two variables. The result can be described in the following way. We associate a matrix C(p) to any polynomial p ∈ C[x, y], where p(x, y) = ∑_{i=0}^{n−1} ∑_{j=0}^{n−1} c_{ij}x^{i}y^{j}. Given a polynomial ψ ∈ C[x] we will denote by Ψ the ideal in C[x, y] generated by (ψ(x), \overline{ψ}(y)) where \overline{ψ}(z) = \overline{ψ(\overline{z})}. The half plane result can be stated in the folloowing way.

Theorem 6.1 If ψ ∈ C[x] and Ψ is the previously defined ideal. Then all zeroes of ψ are in the open right half plane if and only if C((x + y)^{−1}modΨ) is positive definite.

This result can be transformed into other domains as follows. Let r(x, y) ∈ C[x, y] be such that C(r) has rank 2 and signature 0. Then the following is true.

Theorem 6.2 If ϕ ∈ C[x] and Φ is the previously defined ideal. Then all zeroes of ϕ are in the domain Rer(λ, \overline{λ}) > 0 if and only if C(r^{−1}modΦ) is positive definite.

It is worth mentioning that Kalman’s method has been extended, see Djaferis and Mitter [1977], to give a constructive algorithm for the solution of Liapunov related equations.

From my point of view I find the Kalman [1969] approach especially interesting because of the intensive use of modular arithmetic. This leads to a very direct connection to Liapunov’s theorem.

We proceed now to a brief review of the classical approach using quadratic forms. This is the road taken originally by Hermite [1856] and an excellent scholarly exposition of is to be found in Krein and Naimark [1936].

The basic quadratic forms related to the analysis of root location problems are the Hankel and Bezoutian forms. For this set of problems the Bezoutian certainly proves itself the more powerful tool. The reason for this being its linearity properties with respect to its defining polynomials. The Bezoutian B(q, p) of the two polynomials qandp is defined as the quadratic form B(q, p) = (b_{ij}) where

q(z)p(w) − p(z)q(w) = ∑_{i=1}^{n} ∑_{j=1}^{n} b_{ij}z^{i−1}(z − w)w^{j−1}.  (23)

The Bezoutian is a symmetric matrix, linear in its polynomial variables qandp and alternating, i.e. B(p, q) = −B(q, p). For an extensive study of Bezoutians see Helmke and Fuhrmann [1989]. It seems that the most powerful way to study the Bezoutian is the following characterization derived in Fuhrmann [1981a].

Theorem 6.3 Let p, q ∈ F [z], with deg p ≤ deg q. Then the Bezoutian B = B(q, p) of q and p satisfies

B(q, p) = [p(S_{q})]_{co}^{st}.  (24)

Applying Bezoutians, The elementary results concerning root location are given by the following theorem.

Theorem 6.4 Let q be a real polynomial. Then

(i) The number of distinct roots of q is equal to codimKerB(q, q′).

(ii) The number of distinct real roots of q is equal to σ(B(q, q′)).

(iii) All roots of q are real if and only if B(q, q′) ≥ 0.

In the root location problem several Bezoutian related quadratic forms turn out to be useful. In particular let Q = (Q_{jk}), where

Q(z, w) = −i\frac{q(z)q̃(w) − q̃(z)q(w)}{z − w} = ∑_{i=1}^{n} ∑_{j=1}^{n} Q_{jk}z^{j−1}w^{k−1}  (25)

where q̃(z) = ∑_{k=0}^{n} \overline{q}_{k}z^{k} = \overline{q(\overline{z})}. Then the following holds.

Theorem 6.5 Let q be a complex polynomial of degree n. Let

Q = ∑_{j=1}^{n} ∑_{k=1}^{n} Q_{jk}ξ_{j}\overline{ξ}_{k}

be the Hermitian form with the Q_{jk} defined by (25). Then the number of of real zeroes of q together with the number of zeroes of q arising from complex conjugate pairs is equal to δ. There are π more zeroes of q in the upper half plane and ν more in the lower half plane. In particular all zeroes of q are in the upper half plane if and only if Q is positive definite.

Note that the Hermitian form −iB(q, q̃) can also be written as the Bezoutian of two real polynomials, i.e. polynomials with real coefficients. Indeed, let q_{r}, q_{i} denote the real and imaginary parts of q, which are defined by

\begin{matrix}q_{r}(z) = \frac{q(z) + q̃(z)}{2} \\ q_{i}(z) = \frac{q(z) − q̃(z)}{2i} \end{matrix}

we have

−iB(q, q̃) = −iB(q_{r} + iq_{i}, q_{r} − iq_{i})
= −i[B(q_{r}, q_{r}) + iB(q_{i}, q_{r}) − iB(q_{r}, q_{i}) + B(q_{i}, q_{i})]
= 2B(q_{i}, q_{r})

or

− iB(q, q̃) = 2B(q_{i}, q_{r}).

We can arrive at stability results by making a simple change of variable. From the generating function

\frac{q(z)q̃(w) − q̃(−z)q(−w)}{z + w}  (26)

a quadratic form H = (h_{ij}) is induced via

\frac{q(z)q̃(w) − q̃(−z)q(−w)}{z + w} = ∑_{i=1}^{n} ∑_{j=1}^{n} h_{ij}z^{i−1}w^{j−1}.  (27)

The quadratic form associated with the generating function (26) is called the Hermite-Fujiwara form.

Let now q(z) be a real polynomial. We define its even and odd parts, which we denote by q_{+} and q_{−} respectively, by

\begin{matrix}q_{+}(z^{2}) = \frac{q(z) + q(−z)}{2} \\ q_{−}(z^{2}) = \frac{q(z) − q(−z)}{2} \end{matrix}  (28)

Thus we have q(z) = q_{+}(z^{2}) + zq_{−}(z^{2}).

Definition 6.1 Let q(z) be a real monic polynomial of degree m with real simple zeroes

α_{1} < α_{2} < ⋯ < α_{m}

and let p(z) be a real polynomial with positive leading coefficient and zeroes

β_{1} < β_{2} < ⋯ < β_{m−1} if degp = m − 1

and

β_{1} < β_{2} < ⋯ < β_{m} if degp = m.

Then we say that q and p are a real pair if the zeroes satisfy

α_{1} < β_{1} < α_{2} < β_{2} < ⋯ < β_{m−1} < α_{m} if degp = m − 1

and

β_{1} < α_{1} < β_{2} < ⋯ < β_{m} < α_{m} if degp = m

We say that q and p form a positive pair if they form a real pair and α_{m} < 0.

We are ready to give a summary of the Bezoutian related stability criteria. For a proof of the related Hurwitz determinantal conditions in the spirit of this paper we refer to Helmke and Fuhrmann [1989].

Theorem 6.6 Let q(z) = z^{n} + q_{n−1}z^{n−1} + ⋯ + q_{0} be monic of degree n. Then the following statements are equivalent.

(i) q(z) is a stable, or Hurwitz, polynomial.

(ii) The Hermite-Fujiwara form is positive definite.

(iii) The two Bezoutians B(q_{+}, q_{−}) and B(zq_{−}, q_{+}) are positive definite.

(iv) The polynomials q_{+} and q_{−} form a positive pair.

(v) The Bezoutian B(q_{+}, q_{−}) is positive definite and all the coefficients q_{i} are positive.

The equivalence of (i) and (v) is referred to as the Lienard-Chipart theorem. The equivalence of (i) and (ii) is classical and goes back to Hermite, see Krein and Naimark [1936] and Gantmacher [1959]. Also it is well known, see Krein and Naimark [1936], Datta [1978a], Fuhrmann [1981a], that the Hermite-Fujiwara form is isomorphic to the direct sum of the forms B(q_{+}, q_{−}) and B(zq_{−}, q_{+}). So conditions (ii) and (iii) are equivalent. The equivalence of statements (ii) and (iii) is the consequence of the representation of the Hermite-Fujiwara form as a direct sum of two Bezoutians. This is an easy computaion and is worth repeating.

\frac{q(z)q̃(w) − q̃(−z)q(−w)}{(z + w)}

= \frac{(q_{+}(z^{2}) + zq_{−}(z^{2}))(q_{+}(w^{2}) + wq_{−}(w^{2})) − (q_{+}(z^{2}) − zq_{−}(z^{2}))(q_{+}(w^{2}) − wq_{−}(w^{2}))}{z + w}

= 2\frac{zq_{−}(z^{2})q_{+}(w^{2}) − q_{+}(z^{2})wq_{−}(w^{2})}{z + w}

= 2\frac{z^{2}q_{−}(z^{2})q_{+}(w^{2}) − q_{+}(z^{2})w^{2}q_{−}(w^{2})}{z^{2} − w^{2}} + 2zw\frac{q_{+}(z^{2})q_{−}(w^{2}) − q_{−}(z^{2})q_{+}(w^{2})}{z^{2} − w^{2}}

(29)

One form contains only even terms the other only odd ones. So the positive definiteness of the Hermite-Fujiwara form is equivalent to the positive definiteness of the two Bezoutians B(q_{+}, q_{−}) and B(zq_{−}, q_{+}).

Let g be a rational transfer function with real coefficients. The Cauchy index of g, denoted by I_{g}, is defined as the number of jumps of g from −∞ to +∞ minus the number of jumps from +∞ to −∞. The central result concerning the Cauchy index is the possibility of evaluating the Cauchy index of a rational function as the signature of a quadratic form. This result is generally known as the Hermite-Hurwitz theorem. For a proof and a connection to signature symmetric realizations see Fuhrmann [1983].

Theorem 6.7 (Hermite-Hurwitz) Let g = p/q = ∑_{i=1}^{∞} \frac{g_{i}}{z^{i+1}} be a strictly

proper rational function with p and q coprime and deg q = n. Then

I_{g} = σ(H_{n}) = σ(B(q, p))  (30)

where H_{n} denotes the Hankel matrix

H_{n} = ( \begin{matrix}g_{1} & . & . & . & g_{n−1} \\ . & . & . & . & . \\ . & . & . & . & . \\ . & . & . & . & . \\ g_{n−1} & . & . & . & g_{2n−3} \end{matrix} )

As a consequence of Theorem 6.6 (iii) and the Hermite-Hurwitz theorem we can state the following.

Theorem 6.8 Let q be a real monic polynomial, and let q_{+}, q_{−} be defined as in (28). Let the rational function g be defined by

g(z) = \frac{q_{−}(z)}{q_{+}(z)}.

Then

1. g is proper for odd n and strictly proper for even n.

2. g can be expanded in a power series in z^{−1}

g(z) = g_{0} + g_{1}z^{−1} + ⋯

with g_{0} = 0 in case n is even.

3. The polynomial q is stable if and only if the Hankel matrices

H_{n} = ( \begin{matrix}g_{1} & . & . & . & g_{n−1} \\ . & . & . & . & . \\ . & . & . & . & . \\ . & . & . & . & . \\ g_{n−1} & . & . & . & g_{2n−3} \end{matrix} )

and

(σH)_{n} = ( \begin{matrix}g_{2} & . & . & . & g_{n} \\ . & . & . & . & . \\ . & . & . & . & . \\ . & . & . & . & . \\ g_{n} & . & . & . & g_{2n−2} \end{matrix} )

are both positive definite and g_{0} ≥ 0.

Going back to Equation (29) we are led directly to the following result, see Gantmacher [1959] p. 177-178.

Theorem 6.9 Let q be a real monic polynomial of degree n and let

q(z) = q_{+}(z^{2}) + zq_{−}(z^{2})

Then q is a Hurwitz polynomial if and only if for

g(z) = \frac{−zq_{−}(−z^{2})}{q_{+}(−z^{2})}

or

g(z) = \frac{q_{n−1}z^{n−1} − q_{n−3}z^{n−3} + ⋯}{z^{n} − q_{n−2}z^{n−2} + ⋯}

we have I_{g} = n.

\underline{Proof:} It suffices to compute the Bezoutian of −zq_{−}(−z^{2}) and q_{+}(−z^{2}). This we proceed to do

\frac{−q_{+}(−z^{2})wq_{−}(−w^{2}) + zq_{−}(−z^{2})q_{+}(−w^{2})}{z − w}

= \frac{[−q_{+}(−z^{2})wq_{−}(−w^{2}) + zq_{−}(−z^{2})q_{+}(−w^{2})](z + w)}{z^{2} − w^{2}}

= \frac{z^{2}q_{−}(−z^{2})q_{+}(−w^{2}) − q_{+}(−z^{2})w^{2}q_{−}(−w^{2})}{z^{2} − w^{2}}

+zw\frac{q_{−}(−z^{2})q_{+}(−w^{2}) − q_{+}(−z^{2})q_{−}(−w^{2})}{z^{2} − w^{2}}

So the Bezoutian is isomorphic to B(zq_{−}, q_{+}) ⊕ B(q_{+}, q_{−}).

This result has an immediate interpretation in terms of continued fractions. This will be discussed in the next section.

To complete our survey of stability we quote the following result which makes the contact with Liapunov’s theorem via the use of polynomial models.

Theorem 6.10 Assume q is a real polynomial of degree n, having no zeroes on the imaginary axis, and let C_{q} be its companion matrix, defined by

C_{q} = [S_{q}]_{st}^{st} = ( \begin{matrix}0 & . & . & . & −q_{0} \\ 1 & & & & . \\ & . & & & . \\ & & . & & . \\ & & & 1 & −q_{n−1} \end{matrix} ) .  (31)

Then the Hermite-Fujiwara matrix H is a solution of a Liapunov equation

C_{q}H + HC̃_{q} = −Q

with Q a nonnegative definite quadratic form.

7 CONTINUED FRACTIONS

Continued fractions have along history in mathematics. Kalman’s contribution to this subject, Kalman [1979], is by no means the first use of continued fractions in the area of system theory. A variety of ad-hoc methods related to continued fractions and Pade approximationhave been used in system theoretic problems. However Kalman’s contribution seemed the first coherent exposition on the connection of the Euclidean algorithm, continued fractions, the partial realization problem, canonical forms etc. This was a highly influential paper that triggered a lot of other work in this area. The papers by Gragg and Lindquist [1983], Antoulas [1986] and Helmke and Fuhrmann [1989] are just a small sample.

The use of continued fractions as a parametrization for Rat(n) has been suggested in that paper. Some work on the topological aspects of this approach was started by Fuhrmann and Krishnaprasad [1986] and continued by Helmke, Hinrichsen and Manthey [1989]. Also, the idea of using continued fractions for the construction of balanced realizations in Fuhrmann [1989] is along the lines of Kalman [1979] and Ober [1987].

Let g be a strictly proper rational function and let g = p/q be an irreducible representation of g with q monic of degree n. We define, using the division rule for polynomials, a sequence of polynomials q_{i}, and a sequence of nonzero constants β_{i} and monic polynomials a_{i+1}(z), referred to as \underline{atoms} by

\begin{matrix}q_{−1} = q, q_{0} = p \\ q_{i+1}(z) = a_{i+1}(z)q_{i}(z) − β_{i}q_{i−1}(z) \end{matrix}  (32)

with deg q_{i+1} < deg q_{i}. The procedure ends when q_{r} is the g.c.d. of p and q. Since p and q are assumed coprime q_{r} is a nonzero constant.

The atoms a_{1}, …, a_{r} are real monic polynomials of degrees n_{1}, …, n_{r} such that

n_{1} + ⋯ + n_{r} = n  (33)

and β_{i} are nonzero real numbers with

δ_{i} = signβ_{i}.  (34)

In terms of the β_{i} and the a_{i}(z), g has the continued fraction representation

g(z) = \frac{β_{0}}{a_{1}(z) − \frac{β_{1}}{a_{2}(z) − \frac{β_{2}}{a_{3}(z) − ... \frac{β_{r−2}}{a_{r−1}(z) − \frac{β_{r−1}}{a_{r}(z)}}}}}  (35)

Using the linearity properties of the Bezoutian in conjunction with the Euclidean algorithm, or equivalently with the continued fraction expansion (35), we are led to the following result. For a proof see Fuhrmann [1983].

Theorem 7.1 (Frobenius) Let g be rational having the continued fraction representation (35). Then

σ(H_{g}) = ∑_{i=1}^{r}(signΠ_{j=0}^{i−1}β_{j})\frac{1 + (−1)^{dega_{i}−1}}{2}  (36)

The continued fraction representation (35) can be translated immediately into a canonical form realization which incorporates the atoms.

Theorem 7.2 Let the strictly proper transfer function g = \frac{p}{q} have the se-

quence of atoms {a_{i+1}(z), β_{i}}, and assume that n_{1}, …, n_{r} are the degrees of the atoms and

a_{k}(z) = ∑_{i=0}^{n_{k}−1} a_{i}^{(k)}z^{i} + z^{n_{k}}.  (37)

Then g has a realization of the form (A, b, c) of the form

A = ( \begin{matrix}A_{11} & A_{12} \\ A_{21} & A_{22} & . \\ & . & . & . \\ & & . & . & A_{r−1 r} \\ & & & A_{r r−1} & A_{rr} \end{matrix} )  (38)

where

A_{ii} = ( \begin{matrix}0 & . & . & . & −a_{0}^{(i)} \\ 1 & & & & . \\ & . & & & . \\ & & . & & . \\ & & & 1 & −a_{n_{i}−1}^{(i)} \end{matrix} ) , i = 1, … r,  (39)

A_{i+1 i} = ( \begin{matrix}0 & . & . & 0 & 1 \\ . & & & & 0 \\ . & & & & . \\ . & & & & . \\ 0 & . & . & . & 0 \end{matrix} )  (40)

and A_{i i+1} = β_{i−1}A_{i+1 i}.

b = ( \begin{matrix}1 \\ 0 \\ . \\ . \\ 0 \end{matrix} ) , c = ( 0 . . β_{0} 0 . . 0 )  (41)

the β_{0} being in the n_{1} position.

\underline{Proof:} Two sequences of polynomials P_{k} and Q_{k} are defined by the three term recursion formulas

\begin{matrix}P_{−1} = −1, P_{0} = 0 \\ P_{k+1}(z) = a_{k+1}(z)P_{k}(z) − β_{k}P_{k−1}(z) \end{matrix}  (42)

and

\begin{matrix}Q_{−1} = 0, Q_{0} = 1 \\ Q_{k+1}(z) = a_{k+1}(z)Q_{k}(z) − β_{k}Q_{k−1}(z) \end{matrix}  (43)

The expansion of P_{k}/Q_{k} in powers of z^{−1} agrees with that of g up to order 2 ∑_{i=1}^{k} deg a_{i} + deg a_{k+1}, see Gragg and Lindquist [1983]. The set

B_{or} = {1, z, …, z^{n_{1}−1}, Q_{1}, zQ_{1}, …, z^{n_{2}−1}Q_{1}, …, Q_{r−1}, …, z^{n_{r−1}−1}Q_{r−1}}

is clearly a basis for X_{q} as it contains one polynomial for each degree between 0 and n − 1. We call this basis the orthogonal basis because of its relation to orthogonal polynomials. In this connection see Gragg [1974] and Fuhrmann [1988].

To complete the proof we use the shift realization (18) of g which is minimal, by the coprimeness of p and q. Its matrix representation with respect to the basis B_{or} proves the theorem. In the computation of the matrix representation we lean heavily on the recursion formulas (43).

The previous construction can be applied also to obtain an output-feedback canonical form. Indeed, suppose ĝ = p̂/q̂ is output feedback equivalent to g = p/q. Then p̂(z) = αp(z), α ≠ 0 and q̂(z) = q(z) − kp(z). Therefore

q_{1}(z) = a_{1}(z)p(z) − β_{0}q(z)

implies

αq_{1}(z) = a_{1}(z)(αp(z)) − αβ_{0}q(z)
= a_{1}(z)(αp(z)) − αβ_{0}(q̂(z) + kp(z))
= (a_{1}(z) − β_{0}k)(p̂(z)) − (αβ_{0})q̂(z)

and so

β̂_{0} = αβ_{0} â_{1}(z) = a_{1}(z)p(z) − β_{0}k.

However, as q_{1}/p = q̂_{1}/p̂ all other atoms of g and ĝ coincide. Thus, changing the transfer function g(z) by output feedback amounts to rescaling β_{0} by a nonzero constant and to changing the constant part of a_{1}(z) by an arbitrary real number, i.e. a_{1}(z) ↦ a_{1}(z) − σ. In particular the degrees n_{i} of the atoms are output feedback invariant, see Fuhrmann and Krishnaprasad [1986], Helmke and Fuhrmann [1989]. It follows that any transfer function g(z) as in Equation (35) is output feedback equivalent to a unique transfer function ĝ given by

ĝ(z) = \frac{1}{â_{1}(z) − \frac{β_{1}}{a_{2}(z) − \frac{β_{2}}{a_{3}(z) − ... \frac{β_{r−2}}{a_{r−1}(z) − \frac{β_{r−1}}{a_{r}(z)}}}}}  (44)

with â_{1}(z) = a_{1}(z) − a_{1}(0).

The map g ↦ ĝ defines a canonical form for output feedback, and by choosing a state space realization for ĝ as in Kalman [1979] or Gragg and Lindquist [1983], the following theorem is proved.

Theorem 7.3 Let (A, b, c) be a minimal realization of a transfer function g of McMillan degree n. Let a_{i}(z), β_{i} be the sequence of atoms of g and assume that n_{1}, …, n_{r} are the degrees of the atoms and

a_{k}(z) = ∑_{i=0}^{n_{k}−1} a_{i}^{(k)}z^{i} + z^{n_{k}}.  (45)

Then (A, b, c) is output feedback equivalent to a unique system of the form (Â, b̂, ĉ) of the form

b̂ = ( \begin{matrix}1 \\ 0 \\ . \\ . \\ 0 \end{matrix} ) , ĉ = ( 0 . . 1 0 . . 0 )  (46)

the 1 being in the n_{1} position.

Â = ( \begin{matrix}A_{11} & A_{12} \\ A_{21} & A_{22} & . \\ & . & . & . \\ & & . & . & A_{r−1 r} \\ & & & A_{r r−1} & A_{rr} \end{matrix} )  (47)

where

A_{11} = ( \begin{matrix}0 & . & . & . & 0 \\ 1 & & & & −a_{1}^{(1)} \\ & . & & & . \\ & & . & & . \\ & & & 1 & −a_{n_{1}−1}^{(1)} \end{matrix} ) , A_{ii} = ( \begin{matrix}0 & . & . & . & −a_{0}^{(i)} \\ 1 & & & & . \\ & . & & & . \\ & & . & & . \\ & & & 1 & −a_{n_{i}−1}^{(i)} \end{matrix} ) , i = 2, … r,

A_{i+1 i} = ( \begin{matrix}0 & . & . & 0 & 1 \\ . & & & & 0 \\ . & & & & . \\ . & & & & . \\ 0 & . & . & . & 0 \end{matrix} )  (49)

and A_{i i+1} = β_{i−1}A_{i+1 i}.

We complete the paper with a short study of balanced realizations in a special case. We construct such a realization for the case of an asymptotically stable, all-pass transfer function.

To begin, we recall that a minimal realization (A, B, C, D), of an asymptotically stable transfer function g is called balanced if there exists a diagonal matrix diag(σ_{1}, …, σ_{n}) such that

\{ \begin{matrix}AΣ + ΣÃ & = & −BB̃ \\ ÃΣ + ΣA & = & −C̃C. \end{matrix}  (50)

The matrix Σ is called the grammian of the system (A, B, C, D) and its diagonal entries are called the singular values of the system. They are equal to the singular values of an induced Hankel operator, H_{g}, where this Hankel operator acts between the Hardy spaces of the rigt and left half-planes.

Thus let g = \frac{d^{∗}}{d} be an asymptotically stable, all-pass transfer function. This means, see Glover [1984] or Fuhrmann [1989], that d is stable and the singular values satisfy σ_{1} = ⋯ = σ_{n} = 1.

d(z) = d_{+}(z^{2}) + zd_{−}(z^{2}).  (51)
d^{∗}(z) = d_{+}(z^{2}) − zd_{−}(z^{2}).  (52)

Then

g(z) = \frac{d^{∗}(z) − d(z)}{d(z)} + 1 = \frac{−2zd_{−}(z^{2})}{d_{+}(z^{2}) + zd_{−}(z^{2})} + 1.

Next we put

f(z) = \frac{−zd_{−}(z^{2})}{d_{+}(z^{2})},

i.e. f is constant output feedback equivalent to g. Hence, once we get a canonical form for f, the one for g will easily follow. Next we introduce yet another auxiliary strictly proper rational function by defining

h(z) = \frac{−zd_{−}(−z^{2})}{d_{+}(−z^{2})}.

By Theorem 6.9 the Cauchy index I_{h} of h is equal to n = deg d. This piece of information implies two specific representations for h, one additive and the other a continued fraction representation. For the additive representation we must have n real simple poles for h with positive residues. Thus

h(z) = ∑_{k=1}^{n} \frac{γ_{k}^{2}}{z − α_{k}}

with α_{1} < ⋯ < α_{n}. The other representation, and the more interesting one for our purposes, is a continued fraction representation of the form given in Equation (35) i.e.

h(z) = \frac{β_{0}}{a_{1}(z) − \frac{β_{1}}{a_{2}(z) − \frac{β_{2}}{a_{3}(z) − ... \frac{β_{n−2}}{a_{n−1}(z) − \frac{β_{n−1}}{a_{n}(z)}}}}}  (53)

where a_{i}(z), i = 1, …, n are monic of degree one and all the β_{i} are positive.

Theorem 7.4 Let g be an asymptotically stable all-pass function. Then g has a realization of the form

(( \begin{matrix}0 & α_{1} & 0 \\ −α_{1} & . & . \\ & . & . & . \\ & & . & . & α_{n−1} \\ & & & −α_{n−1} & −k \end{matrix} ) , ( \begin{matrix}0 \\ . \\ . \\ . \\ α_{0} \end{matrix} ) , ( 0 … α_{0} )) (54)

\underline{Proof:} We will distinguish between two cases.

CASE 1 Assume that n = deg d is even. Therefore the function h has an even denominator and an odd numerator. Thus we can be more explicit as far as the continued fraction is concerned. Indeed, put q_{−1}(z) = d_{+}(−z^{2}) and q_{0}(z) = −zd_{−}(−z^{2}) and set up the Euclidean algorithm with the recursion

q_{i+1}(z) = a_{i}(z)q_{i}(z) − β_{i−1}q_{i−1}(z)  (55)

with the β_{i} chosen so that the a_{i} are monic and deg q_{i+1} < deg q_{i} < deg q_{i−1}. Let a_{i}(z) = z−α_{i}, i = 1, …, n. We will show by induction that the sequence of polynomials is alternatingly even and odd. In fact the initialization ensures q_{−1} to be even and q_{0} to be odd. So assume parity alternates for all indices till i. From equation (55) it follows that

\frac{1}{q_{i−1}(z)} = \frac{β_{i}}{a_{i}(z)q_{i}(z) − q_{i+1}(z)}  (56)

and hence

\frac{q_{i}(z)}{q_{i−1}(z)} = \frac{β_{i}}{a_{i}(z) − \frac{q_{i+1}(z)}{q_{i}(z)}} .  (57)

Now \frac{q_{i}(z)}{q_{i−1}(z)} is odd by the induction hypothesis. So a_{i}(z) − \frac{q_{i+1}(z)}{q_{i}(z)} is also

odd. This forces both a_{i}(z) and \frac{q_{i+1}(z)}{q_{i}(z)} both to be odd. Thus a_{i}(z) = z and

q_{i+1} has opposite parity to q_{i}. Thus h has the representation

h(z) = \frac{α_{0}^{2}}{z − \frac{α_{1}^{2}}{z − \frac{α_{2}^{2}}{z − ... \frac{α_{n−2}^{2}}{z − \frac{α_{n−1}^{2}}{z}}}}}  (58)

By Theorem 9.4 in Helmke and Fuhrmann [1989] h(z) has a realization of the form

(( \begin{matrix}0 & α_{1}^{2} & 0 \\ 1 & . & . \\ & . & . & . \\ & & . & . & α_{n−1}^{2} \\ & & & 1 & 0 \end{matrix} ) , ( \begin{matrix}0 \\ . \\ . \\ . \\ α_{0} \end{matrix} ) , ( 0 … α_{0} ))

which is similar, through the matrix diag(ρ_{0}, …, ρ_{n−1}) with ρ_{0} = 1 and \frac{ρ_{i+1}}{ρ_{i}} = α_{i}^{2}, to the matrix

(( \begin{matrix}0 & α_{1} & 0 \\ α_{1} & . & . \\ & . & . & . \\ & & . & . & α_{n−1} \\ & & & α_{n−1} & 0 \end{matrix} ) , ( \begin{matrix}0 \\ . \\ . \\ . \\ α_{0} \end{matrix} ) , ( 0 … α_{0} )) .

To go back from h to f we note that

f(z) = −ih(iz) = −i(−iz)\frac{d_{−}(z^{2})}{d_{+}(z^{2})}.

So f has only simple imaginary axis zeroes.

From the continued fraction expansion (58) we get

f(z) = \frac{−iα_{0}^{2}}{iz − \frac{α_{1}^{2}}{iz − \frac{α^{2}}{iz − ... \frac{α_{n−2}^{2}}{iz − \frac{α_{n−1}^{2}}{iz}}}}} = \frac{−α_{0}^{2}}{z + \frac{α_{1}^{2}}{z + \frac{α^{2}}{z + ... \frac{α_{n−2}^{2}}{z + \frac{α_{n−1}^{2}}{z}}}}} = (59)

Using again the continued fraction canonical form we get a realizaton for f given by

(( \begin{matrix}0 & α_{1}^{2} & 0 \\ −1 & . & . \\ & . & . & . \\ & & . & . & α_{n−1}^{2} \\ & & & −1 & 0 \end{matrix} ) , ( \begin{matrix}0 \\ . \\ . \\ . \\ α_{0} \end{matrix} ) , ( 0 … α_{0} ))

and by similarity to

(( \begin{matrix}0 & α_{1} & 0 \\ −α_{1} & . & . \\ & . & . & . \\ & & . & . & α_{n−1} \\ & & & −α_{n−1} & 0 \end{matrix} ) , ( \begin{matrix}0 \\ . \\ . \\ . \\ α_{0} \end{matrix} ) , ( 0 … α_{0} ))

Now g is obtained back from f by the inverse output-feedback transformation. Thus g is realized by

(( \begin{matrix}0 & α_{1} & 0 \\ −α_{1} & . & . \\ & . & . & . \\ & & . & . & α_{n−1} \\ & & & −α_{n−1} & −k \end{matrix} ) , ( \begin{matrix}0 \\ . \\ . \\ . \\ α_{0} \end{matrix} ) , ( 0 … α_{0} )) (60)

with k = \frac{α_{0}^{2}}{2} .

Clearly the realization in (60) is balanced with Σ = I.

CASE 2 This case can be treated similarly, however the details will be skipped.

We note in passing that the above realization is a special case of Darlington synthesis, see Brockett [1970] and Krishnaprasad [1980].

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