Causal Factorization and Linear Feedback

An algebraic framework for the investigation of linear dynamic output feedback is introduced. Pivotal in the present theory is the problem of causal factorization, i.e. the problem of factoring two systems over each other through a causal factor. The basic issues are resolved with the aid of the new concept of latency kernels.


INTRODUCTION.
In recent years the system theory literature has seen a rapidly growing interest in questions associated with linear feedback. In the early 1960 's, linear control theory centered chiefly around quadratic (Gaussian) optimal problems and the resulting feedback designs. Later, interest in feedback shifted to a variety of so-called "synthesis" problems. These included the well-known problem of observer design (see Luenberger [1966] ), the pole shifting theorem and related issues (Wonham [1967] Simon and Mitter [1968], Brash and Pearson [1970], Heymann [1968]) as well as the decoupling problem (Falb and Wolovich [1967], Gilbert [1969], Wonham and Morse [1970], Morse and Wonham [1970] ). All of these feedback synthesis problems, as well as many others, were formulated and resolved within the framework of state space representations. While most of the work was done with the use of conventional state equations, the work of Wonham and Morse was distinguished by its "coordinate free" setting and initiated what later developed into the celebrated "geometric theory" of linear control (see, e.g., Wonham [1979]).
The current growing interest in linear feedback differs significantly from that of the past both in character and in its source of motivation. While previously the study of feedback was largely oriented at problem solving, the current interest is motivated by a desire of gaining insight into the general nature of linear feedback-chiefly from an algebraic point of view. Much of the motivation for the present trend can be traced back to the work of Rosenbrock [1970] , in which polynomial matrix techniques were used for the study of a variety of (linear) control theoretic questions. Particularly useful turned out to be techniques based on polynomial fraction representations of transfer functions (see, e.g., Heymann [1972], Wolovich [1974], Forney [1975], Fuhrmann [1976]). In this setting of fraction representations, feedback was first studied in Heymann [1972] (see especially Chapter 6 therein), and in a polynomial module framework the study of feedback was initiated by Eckberg [1974]. State feedback also received attention in an algebraic framework by Morse [1975]. A different approach to the study of linear feedback was taken in Hautus and Heymann [1978], where the fundamental underlying object was taken to be the input-output map of the system. There, static linear state feedback was investigated in an algebraic framework consistent with the setting of the (classical) module theory of linear realization as introduced by Kalman (see, e.g., Kalman et al. [1969, Chapter 10] ). More recently, state feedback was also examined in Fuhrmann [1979] using what he termed "polynomial models", and in Münzner and Prätzel-Wolters (1979a], [1979b], [1979c] in a module and category theoretic framework. While these various approaches to the study of feedback differ from each other substantially both in the underlying concept and in philosophy, they commonly converge on essentially the same (standard) issues that characterize state feedback. It is significant, however, that no success (and, in fact, very little effort, if any) has been reported in respect to output, as opposed to state feedback. When various fundamental questions in regard to output feedback are examined, it becomes immediately clear that difficulties arise that are completely absent in the state-feedback setting. In fact, one discovers immediately that crucial insight is missing. It turns out that the chief reason for this state of affairs is the fact that all of the presently existing algebraic theory of linear systems, and especially that of feedback, rests in one way or another on the theory of modules over the ring [ ] of polynomials and on polynomial matrices. This algebraic machinery is completely satisfactory to develop a fairly comprehensive framework for state feedback. It is not adequate, though, to deal with output-feedback where issues associated with causality become significantly more intricate. †The latex version of this paper was obtained from the original by using optical character recognition at Mathpix.com. We express our profound thanks to Mathpix. The present paper deals in a comprehensive way with the problem of causal output feedback. A related question which receives a great deal of attention in the paper and on which much of the theory hinges is the so-called causal factorization problem. This is the problem of when a given linear input-output map can be factored over another one by a causal linear map. Through the resolution of this issue, questions associated with dynamic causal output feedback are then also resolved. Attention is also given to the static factorization problem as well as the problem of static feedback where special emphasis is placed on the state-feedback case.
A crucial role in the present theory is played by the newly introduced concept of latency. In the discrete time setting, latency expresses "degree of causality" and (intuitively) refers to the intrinsic delay which inputs encounter before output responses are produced. Latency is algebraically expressed by modules over the ring −1 of power series (in −1 over a field ). These modules arise in a natural way when the concept of causality is studied algebraically and in fact are readily seen to be the natural algebraic device for the study of feedback. The paper is organized as follows. In §2 the basic concepts of Λ -linear maps causality, linear i/o maps as well as linear i/s maps, which have been investigated in detail in Hautus and Heymann [1978], are reviewed. The conceptual viewpoint, on which the present investigation of feedback rests, is discussed in §3. An important technical concept that arises in the algebraic study of linear systems both in connection with the [ ] -module theory and the −1 -module theory is that of "proper bases" and "proper independence". This is the topic of $4. Section 5 is devoted to the investigation of causal factorization, the main result being Theorem 5.2 and its corollaries. Results are also obtained on static feedback (Theorems 5.10 and 5.14 ). In §6 the problem of invariants is investigated in detail and explicit characterizations are derived and exhibited. The role of the latency kernels and latency indices is also discussed. The paper is concluded in $7 with an investigation of the interesting question of feedback (design) limitations. It is shown that the essential limitation to the possibility of causal feedback implementation of precompensators is the system's latency. In particular, precompensators can be implemented as causal feedback devices modulo a "precompensator remainder" whose dynamic order need not exceed the sum of the system's latency indices.

ΛK -LINEAR MAPS, CAUSALITY AND INPUT-OUTPUT BEHAVIOR.
We shall adopt a terminology and setup consistent with that of Hautus and Heymann [1978]. Let be a field and let be a -linear space. The class of all truncated -valued Laurent series of the form of -valued Laurent series is endowed with a field structure under the operation of convolution as multiplication and coefficientwise addition. In particular, for = ∑ ∞ = 0 − and ′ = ∑ ∞ = ′ − in Λ , the product ′ is given by and the sum + ′ is given by With Λ as the underlying field it then follows that, with convolution as the scalar multiplication and with the usual coefficientwise addition, the set Λ becomes a Λ -linear space. When is a finite dimensional -linear space, say of dimension then so is Λ as a Λ -linear space. It is readily observed that; under the same operations of convolution as multiplication and coefficientwise addition, the field Λ contains (as subobjects) also (i) the ring [ ], or in our notation Ω + , of polynomials in ; (ii) the ring −1 , or in our notation Ω − , of formal power series in −1 ; and finally, (iii) the field itself. It, thus, follows immediately that the set Λ is not only a Λ -linear space but is simultaneously also an Ω + -module, an Ω − -module and a -linear space As we shall see, these facts turn out to be of central importance in the theory. Now, we let ℤ denote the integers and for an element ∈ Λ , given by (2.1), we define the order of by If ≠ 0 and 0 = ord , we call the coefficient 0 the leading coefficient of . Let and be -linear spaces. We shall call the input value space and the output value space of an underlying linear system Σ. The Λ -linear spaces Λ and Λ are then called the extended input space and extended output space, respectively Elements = Σ − ∈ Λ and = Σ − ∈ Λ , called, respectively, (extended) inputs and (extended) outputs, are identified with time sequences and (with being identified as time marker). Let̄ ∶ Λ → Λ be a -linear map. We say that̄ is time invariant if ( ⋅ ) = ⋅̄ ( ) for all ∈ Λ , so that̄ is time invariant whenever it is a Λ -linear map (Wyman [1972].. Next, for a Λ -linear map̄ ∶ Λ → Λ we define the order of̄ by If the map̄ is the zero map then ord̄ ∶= ∞; otherwise ord̄ < ∞. While it is possible that ord̄ = −∞ we shall not concern ourselves here with this case and confine our attention to maps of finite order. This is clearly always the case when (and hence also Λ ) is finite dimensional.
We now introduce the following (see also Hautus and Heymann [1978]).
Let denote the -linear space of -linear maps → and let Λ denote the Λ -linear space of all -Laurent series. We identify this space with the space of Λ -linear maps Λ → Λ of finite order as follows. We define the -linear maps and with every Λ -linear map̄ ∶ Λ → Λ we associate the Laurent series where, for each ∈ ℤ (2.7) ∶= (̄ ) ∶=̄ ⋅̄ ⋅̄ The Laurent series (2.6) is called the impulse response or the transfer function of̄ . If = Σ − ∈ Λ is any element, then the action of̄ on is given by It is thus immediately seen that (2.9) ord̄ = min | (̄ ) ≠ 0 , whence we have the following characterization of causality in terms of the transfer function: The map̄ is causal if and only if (̄ ) = 0 for < 0 and strictly causal if and only if (̄ ) = 0 for ≦ 0. We also have the following easily verified proposition.
We associate with an extended linear i/o map̄ a restricted linear i/ o map̃ which is obtained as follows (see also Hautus and Heymann [1978]). Inputs are restricted to the subset Ω + ⊂ Λ , called the restricted input space, and consist of all inputs that terminate at = 0, i.e., elements of the form Σ ≤0 − . Outputs are observed only for ≧ 1, that is, in the subset −1 Ω − which is, of course, in bijective correspondence with the Ω + -quotient module Γ + ∶= Λ ∕Ω + which we call the restricted output space. The restricted linear i/o map̃ ∶ Ω + → Γ + associated with̄ is then defined bỹ = + ⋅̄ ⋅ + where + ∶ Ω + → Λ is the canonical injection and + ∶ Λ → Γ + is the canonical projection. Clearly, since + and + are Ω + -module homomorphisms, so is alsõ and we have the following: Definition. 2.12. A map̃ ∶ Ω + → Γ + is called a restricted linear i/o map if it is an Ω + -module homomorphism.
Next, we define the linear output response (or output value) map ∶ Ω + → associated with a given linear i/o map̄ (or̃ ) as follows: is called reachable if the associated output value map is surjective. If ∶ Ω + → is any -linear map, it can be regarded as an output value map of a linear system. In particular, the restricted and extended linear i/o maps associated with are then given by The output value map , which gives for each (restricted) input the value of the output at time = 1, is clearly a -linear map. In some special cases, there exists an Ω + -module structure on , compatible with its -vector space structure, such that the output value map is not just -linear but is also an Ω + -module homomorphism. When this is the case, then for each ∈ Ω + and for each positive integer , = ( ), whence, by (2.15), knowledge of the output value at time = 1 implies knowledge of the whole ensuing output sequence. This is therefore precisely the case when the system's output "qualifies" as state, a fact which motivates the following definition (for greater detail the reader is referred to Hautus and Heymann [1978]): Definition. 2.17. An extended linear i/o map̄ ∶ Λ → Λ is called an extended linear input-state (or i/s) map if there exists an Ω + -module structure on , compatible with its -linear structure, such that the output value map =̄ 1 ⋅̄ ⋅ + is an Ω + homomorphism. The associated restricted map̃ is called restricted linear i/s map.
If and are -linear spaces and ∶ → is a -linear map, then it induces in a natural way a Λ -linear map which we call static as follows: In a similar way induces also static Ω + and Ω − -homomorphisms. We shall need the following characterizations of linear i/s maps, from Hautus and Heymann [1978] Theorem.

FEEDBACK AND CAUSAL FACTORIZATION-GENERAL CONSIDERATIONS.
We shall be concerned with the setup described by the block diagram in Fig. 3.1. Herē ∶ Λ → Λ is an extended linear i/o map, called the open loop system,̄ ∶ Λ → Λ is a causal Λ -linear map called the (output) feedback compensator,̄ pr ∶ Λ → Λ is a Λ -linear bicausal isomorphism called (bicausal) precompensator and̄ po ∶ Λ → Λ is a Λ -linear bicausal isomorphism called (bicausal) postcompensator. In case any of the maps̄ ,̄ pr or̄ po is static we shall call it, respectively a static feedback, pre or post compensator. Now, since the map̄ is causal and̄ is strictly causal, it readily follows that the composite maps̄ ⋅̄ ∶ Λ → Λ and̄ ⋅̄ ∶ Λ → Λ are both strictly causal. Letting denote both of the corresponding identity maps, we see that both of the maps ( +̄ ̄ ) ∶ Λ → Λ and ( +̄ ̄ ) ∶ Λ → Λ are bicausal isomorphisms. It follows that the setup of Fig. 3.1 is "well-posed" in the sense that there is a strictly causal Λ -linear map Λ → Λ ∶ ↦ given by either of the following composite maps: Using again block diagrams, (3.1) and (3.2) can be described, respectively, as in Fig. 3.2a and 3.2b. In both descriptions, the dashed blocks represent bicausal mappings, so that the compensator configuration of Fig.  3.1 can always be represented equivalently Because of the obvious duality between the precompensator situation and the postcompensator situation, there is no need to discuss both of them in detail. since practical interest in postcompensators is at best limited, we shall henceforth confine our attention to precompensation, and discuss postcompensators only in connection with certain mathematical questions.
For various reasons, not to be elaborated on here, feedback compensation is preferred over external compensation whenever possible. Thus, one is interested in the following problem.
where 0 is a static Λ -linear map and h −1 is the transfer function of a strictly causal maph ∶ Λ → Λ representing the strictly causal part of̄ −1 . Hence we can always decompose the map̄ −1 as −1 = +h with static andh strictly causal. The causal feedback problem 3.3 is therefore essentially equivalent to the following.
Causal factorization problem 3.4. Let̄ ∶ Λ → Λ be a given strictly causal Λ -linear map. (a) Under what conditions can a strictly causal Λ -linear maph ∶ Λ → Λ be factored causally over̄ , i.e., when does there exist a causal map̄ ∶ Λ → Λ such thath =̄ ⋅̄ ? (b) Under what conditions can every strictly causal Λ -linear maph ∶ Λ → Λ be factored causally over̄ ? It is readily noted that the strict causality of the maps̄ andh is inessential to the causal factorization problem, and arises in problem 3.4 only because of the specific requirements of the feedback problem. Indeed, ifh factors causally over̄ , i.e., if there exists a causal̄ such thath =̄ ⋅̄ , then for each integer we also have h = ̄ ̄ =̄ ⋅ ̄ so that ℎh factors causally over ̄ , and for sufficiently large positive (unlessh or̄ are zero) the maps h and ̄ are not causal. Thus, the causal factorization problem can be stated in the following less restrictive way: Given two Λ -linear maps̄ ∶ Λ → Λ andh ∶ Λ → Λ (where , and are -linear spaces ), when does there exist a causal Λ -linear map̄ ∶ Λ → Λ such that the following diagram in Fig. 3.3 commutes FIG If the causality requirement of̄ is dropped, the factorization problem is standard (see, e.g., Greub [1967]) and ℎ factors over̄ if and only if ker̄ ⊂ kerh. Yet this condition does not say anything about the causality of̄ . To deal efficiently with the causality issue, we reintroduce the concept of causality using an approach which is algebraically more tractable.
Let̄ ∶ Λ → Λ be a Λ -linear map. We can characterize causality of̄ as follows (compare with our definitions of causality in §2 ): (3.5) The map̄ is causal if and only if ∈ Ω − implies̄ ( ) ∈ Ω − . Similarly, we have: (3.6) The map̄ is strictly causal if and only if ∈ Ω − implies̄ ( ) ∈ Ω − . Let us denote the Ω − -quotient module Λ ∕Ω − by Γ − , and let − ∶ Λ → Γ − denote the canonical projection. The following can then be easily verified by the reader. We shall use the characterizations of the above proposition extensively in the following sections.

PROPER INDEPENDENCE AND PROPER BASES.
Let be a field and let ∶= . For an element 0 ≠ ∈ Λ , denote bŷ the leading coefficient of . If = 0 we shall say that̂ = 0.
Below we derive a variety of properties of properly independent sets, of proper bases and of proper direct sum decompositions. Our objective is to develop this theory here only to the extent required in the sequel. Many further results have been omitted, and the reader can, for example, easily verify that the converses of a number of our results are also valid. A more extensive exposition of this and related topics will be published elsewhere Lemma. 4.2. If 1 , ⋯ , ∈ Λ is a properly independent set of vectors, then (i) it is Λ -linearly independent, and (ii) for every set of scalars 1 , ⋯ , ∈ Λ the following holds Proof. We shall prove the lemma by showing that if either (i) or (ii) fails to hold then the set 1 , ⋯ , is not properly independent. If 1 , ⋯ , ∈ Λ is any set of scalars then, by definition, ord ∑ =1 ≧ ∶= min ord | = 1, ⋯ , . If either (i) or (ii) fails to hold, there exist 1 , ⋯ , ∈ Λ , not all zero, such that either ∑ and consider the terms of order in Σ . This yields ∑ =1 ̂ ̂ = 0, implying that̂ 1 , ⋯ ,̂ are -linearly dependent since not all the ̂ are zero. Hence 1 ⋯ , are not properly independent, completing the proof. □ The condition of Lemma 4.2( ii ) has been called the "predictable degree property in Forney [1975], in the (analogous) setting of "minimal polynomial bases" for rational vector spaces. We shall adopt this terminology and call the property of Lemma 4.2(ii) the predictable order property.
To avoid possible confusion in the ensuing discussion where we shall deal with both -linear and Λ -linear spaces, we shall use subscripts to emphasize the field. Thus, for example, span Λ 1 , ⋯ , denotes the Λ -linear subspace spanned by 1 , ⋯ , ∈ Λ , whereas span ̂ 1 , ⋯ ,̂ denotes the -linear subspace spanned bŷ 1 , ⋯ ,̂ ∈ . Similarly, dim Λ ℛ denotes the dimension of a subspace ℛ ⊂ Λ as a ΛK -linear space (to distinguish from -linear). We next have the following theorem.
Theorem. 4.4. Every nonzero Λ -linear subspace ℛ ⊂ Λ has a proper basis. Moreover, every properly independent subset of ℛ can be extended to a proper basis.
Finally, we also have the following variant of the predictable order property.

CAUSAL FACTORIZATION.
We turn now to the causal factorization problem (3.4). As we mentioned earlier, there is no essential need, in characterizing causal factorizability, to assume strict causality, or even causality, of the maps under consideration. We shall therefore begin with the general case and turn to specific consideration of i/o maps later on. We shall assume that the spaces and are finite dimensional, in particular that = and = . For convenience of notation, we shall temporarily use the notation Λ and Λ also in connection with Λ -linear maps̄ ∶ Λ → Λ that are not necessarily i/o maps (i.e., are not necessarily strictly causal).
Let̄ ∶ Λ → Λ be a Λ -linear map and let − ∶ Λ → Γ − ∶= Λ ∕Ω − be the canonical projection. since Ω − is an Ω − -module, so is the quotient Λ ∕Ω − . Thus the map − is an Ω − -homomorphism and so is also the composite −̄ . We have Next we have the following central theorem.

Corollary. 5.4. Let̄ ∶ Λ → Λ be an extended linear i/o map. Then̄ is nonlatent if and only if for every strictly
Let̄ ∶ Λ → Λ be an extended linear i/o map and let̄ ∶ Λ → Λ be a bicausal isomorphism, i.e., a bicausal precompensator for ⃗ . Leth be the strictly causal part of̄ −1 , i.e.,̄ −1 = +h where is static. As we have seen in §3,̄ can be realized as feedback around̄ ifh factors causally over̄ . Theorem 5.2 tells us essentially that the only barrier to realizing a bicausal precompensator as feedback is the relative latency of̄ andh. Corollary 5.4 characterizes the class of i/o maps over which every bicausal precompensator can be realized as feedback. These i/o maps are, as we have seen, the nonlatent maps (a fact which motivated our choice of terminology). Now, a very special and important class of nonlatent maps is that of injective i/s maps. This fact is proved in the following theorem.
Proof. By strict causality of̄ we have that Ω − ⊂ ker −̄ , so that to prove nonlatency we need only to show that ker −1̄ ⊂ Ω − . Let ∈ ker −̄ so that̄ ( ) ∈ Ω − . Write = + + − , where + ∈ 2 Ω + and − ∈ Ω − . The proof will be completed by showing that + = 0 so that ∈ Ω − as claimed. Note that̄ ( − ) ∈ Ω − by the strict causality of̄ so that, in view of the fact that̄ ( ) =̄ + +̄ ( − ) , it follows that̄ + ∈ Ω − . By (2.16) we havē so that, in particular, that −2 + ∈ ker = ker̃ (the equality being a consequence of the i/s property (2.20) ). It follows that̄ −2 + ∈ Ω + , or alternatively, that̄ + ∈ 2 Ω + . since 2 Ω + ∩ Ω − = 0, we conclude that̄ + = 0 or that + = 0 by the injectivity of̄ . □ While Theorem 5.5 deals only with injective i/s maps, it is important to observe that this is not a serious restriction. Indeed, it is shown in Proposition 5.6 below that in the special case of i/s maps (in contrast to i/o maps in general), the kernel is "static"; i.e., if̄ is a noninjective i/s map, then ker̄ = Λ 0 where 0 ⊂ is a subspace. This means that the whole degeneracy lies in the input value space which has been chosen too large, and by restricting the input value space to a proper summand of 0 in , the injectivity is restored.
Proof. Let ∶ → Ω + ∶ ↦ be the canonical injection and define the subspace 0 ⊂ as 0 ∶= ker ⋅ , where is the output value map associated with̄ . sincē is an i/s map we have ker ⋅ = ker̃ ⋅ = ker̄ ⋅̄ with the last equality holding by the strict causality of̄ . Thus̄ 0 ⊂ ker̄ , and since ker̄ is a Λ -linear space we conclude that Λ 0 ⊂ ker̄ . To prove that ker̄ ⊂ Λ 0 , it suffices to prove that if 0 ≠ = ∑ ∞ = 0 − ∈ ker̄ then 0 ∈ 0 . By recursive application of the same argument this will then imply that ∈ 0 for all ≧ 0 . Now by formula (2.16) we have + = 0 for all ∈ ℤ, and since + 0 = 0 the results follow. □ The importance of Theorem 5.5 lies in the fact that it tells us that bicausal precompensation is equivalent, in the sense of solvability, to dynamic state feedback Let̄ ∶ Λ → Λ be an extended linear i/o map. We write (see Hautus and Heymann [1978])̄ = ⋅̄ , where is a static output map and̄ is a reachable i/s map. If̄ is injective (which is always the case when ker̄ does not contain a subspace of the form Λ , 0 ≠ ⊂ ), then every bicausal precompensator can be realized as feedback around̄ . That is, we can write every bicausal̄ ∶ Λ → Λ as̄ −1 = +̄ ̄ , wherē ∶ Λ → Λ is a causal Λ -linear map and is static.

Corollary
To see the bicausality of̄ it suffices to show that it is order preserving. Indeed, let = 1 + 2 ∈ Λ be any element with 1 ∈ Im̄ 1 and 2 ∈ ℛ 1 . Then̄ =̄ 1 1 +̄ 2 2 and using Corollary 4.8 together with the fact that Im̄ 1 and Im̄ 2 form a proper direct sum, we have that ord̄ = min ord̄ 1 1 , ord̄ 2 2 = min ord 1 , ord 2 , where the last equality follows from the order preserving property of̄ 1 and̄ 2 . Using Corollary 4.8 again, together with the fact that Im̄ 1 and ℛ 1 form a proper direct sum, gives that min ord 1 , ord 2 = ord whence ord̄ = ord as claimed and the proof is complete. □ Clearly, the bicausal Λ -linear map̄ of Corollary 5.7 can be regarded as a bicausal postcompensator for̄ 1 , and there is a kind of duality between feedback and compensation which deserves some further comments.
Let̄ ∶ Λ → Λ be an extended linear i∕o map and let̄ pr ∶ Λ → Λ be a bicausal precompensator for̄ . If ∶ Λ → Λ is the strictly causal part of̄ pr , then the causal feedback problem is that of existence of a causal Λ -linear map̄ ∶ Λ → Λ such that̄ =̄ ⋅̄ . The map̄ can be regarded essentially as a causal (but not necessarily bicausal) postcompensator for̄ . Conversely, if̄ po ∶ Λ → Λ is a bicausal postcompensator and if̄ ∶ Λ → Λ the strictly causal part of̄ po , the dual of the above causal factorization problem is that of the existence of a causal Λ -linear map̄ ∶ Λ → Λ such that̄ =̄ ⋅̄ . Herē − can be viewed as a causal, but again not necessarily bicausal, precompensator for̄ . Thus the pre-and postcompensator problems become interrelated through feedback. We can also write down the dual of Corollary 5.7 regarding the problem of bicausal precompensation.
In Hautus and Heymann [1978], the static state feedback problem was investigated. This is the following problem: Given an extended linear i/s map̄ ∶ Λ → Λ under what conditions can a bicausal precompensator̄ ∶ Λ → Λ be written as̄ −1 = + ̄ , where and are static maps. It was shown there that a necessary and sufficient condition for the static state feedback problem to have a solution is that (5.9)̄ −1 (ker̃ ) ⊂ Ω + , wherẽ ∶ Ω + → Γ + is the restricted i/s map associated with̄ . We now turn to the more general question of static output (rather than state) feedback. As we have been doing throughout this paper, we focus our attention on the static factorization problem which is characterized in the following Theorem. 5.10. Let̄ ∶ Λ → Λ andh ∶ Λ → Λ be Λ -linear maps. There exists a static Λ -linear map ∶ Λ → Λ such thath = ⋅̄ if and only if ker̄ 1 ⋅̄ ⊂ ker̄ 1 ⋅h.

FACTORIZATION INVARIANTS-EXPLICIT CALCULATION.
Throughout this section we shall assume that = and = , and we shall study properties of Λ as an Ω − -module as well as properties of submodules thereof The ring Ω − is of course a principal ideal domain, and clearly also a Euclidean domain. The units of Ω − are precisely those elements whose order is zero and each element 0 ≠ ∈ Ω − can be expressed as where 0 ∈ Ω − is a unit. It is clear, therefore, that all the ideals of Ω − are of the form − , forming a chain with −1 being the unique maximal ideal and the only prime. Thus, the ring Ω − is also a local ring and Ω − ∕ −1 is a field, isomorphic to the field 0 which consists of the units of Ω − augmented by zero. We shall make use of the special properties of the ring Ω − in the ensuing discussion.
For a fixed integer , consider the subset − Ω − ⊂ Λ . Clearly, this subset is an Ω − submodule of Λ . Moreover, while Λ itself is not a finitely generated Ω − module, the submodule − Ω − is (and hence is a free module). In fact, it is readily noted that rank Ω − − Ω − = dim Λ Λ = dim . Indeed, if 1 , ⋯ , is a basis for (as well as for Λ ), then − 1 , ⋯ , − is a basis (i.e., a free generator) for − Ω − . Let 0 ≠ Δ ⊂ Λ be an Ω − -submodule. We say that Δ is of finite order if there exists a finite integer such that Δ ⊂ − Ω − . The maximal integer for which the above holds, and which is the least order of elements in Δ, is denoted Δ and is called the order of Δ. We define the order of the zero module as infinity. We have the following: Proposition. 6.1. Let 0 ≠ Δ ⊂ Λ be an Ω − -submodule. Then Δ is finitely generated if and only if it has finite order.
Consider now the special case when = (that is, = ) and let be a nonsingular × matrix with entries in Λ . Then defines, as above, an Ω − homomorphism Ω − → Λ and also (when simply regarded as a transfer function) a Δ -linear map Λ → Λ . Denoting both maps by the same symbol , it is readily verified that the diagram in Fig. 6.1 is commutative,   FIGURE 0.5. FIG. 6.1 where − denotes the canonical injection. since the matrix is nonsingular, the Λ -linear map is invertible. We shall say that the matrix is bicausal if the associated Λ -linear map is bicausal, i.e., if the entries of are in Ω − and its determinant is a unit in this ring (that is, has order zero). In analogy we shall say that a matrix is strictly causal or causal if so is the associated Λ -linear map. Finally, an Ω − -submodule Δ = Ω − ⊂ Λ is called a full submodule if rank Δ = , i.e., if the matrix is nonsingular.
The proof of Theorem 6.2 is elementary and will be omitted. The following corollary will be useful in the sequel.

. Then Δ ⊆ Δ 1 if and only if is causal with equality if and only if is bicausal.
Let Δ ⊂ Λ be a finitely generated Ω − -submodule of rank and order Δ . Then for all integers ≦ Δ , Δ ⊂ − Ω − and for each integer ≧ Δ we define the submodule Δ ⊂ Δ by Clearly − Ω − ⊂ − Ω − for all ≧ , and it follows that (6.5) Δ = Δ Δ ⊃ Δ 2 +1 ⊃ ⋯ > Δ ⊃ Δ +1 ⋯ As an immediate consequence of the fact that if ∈ Δ then −1 ∈ Δ +1 , it is clear that rank Δ = rank Δ for all and the quotient modules (6.6) ∶= Δ ∕Δ +1 are all torsion modules with −1 as annihilators, that is, for each and for each [ ] ∈ −1 [ ] = 0. Next we shall show that the sequence of quotient modules is isomorphic to a chain of (finite dimensional) -linear subspaces of , that is, each is isomorphic to a subspace ⊂ and It is also clear that is injective, since ker = Δ +1 = [0]. Now, for each integer we define ∶= Im . Clearly is then -linearly isomorphic to and ⊂ +1 with Δ − −1 = 0 for all ≧ 0. Also, by the finite dimensionality of , there exists an integer Δ 2 ≧ Δ such that Δ −1 ≠ Δ and Δ + = Δ for all ≧ 0. We call the chain the orderchain of Δ, and the sequence of integers , ∶= dim , we call the order list of Δ In the special case when Δ = ker −̄ wherē is a linear i/o map, we refer to the order chain and the order list of Δ, respectively, also as the latency chain and latency list of̄ .
It is interesting to observe that the integer Δ is also the least integer satisfying the condition that −1 Δ = Δ +1 for all ≧ Δ . Indeed, we have seen that −1 ∕ ⊂ Δ +1 for all j. To see that −1 Δ ⊃ Δ +1 if and only if ≧ Δ , let = ∑ ∞ = +1 − ∈ Δ +1 be any element. Then we can write and clearly ∈ −1 Δ if and only if ′ ∈ Δ , This can hold for every ∈ Δ +1 only if +1 = , whence the necessity that ≧ Δ . The sufficiency of the condition is an immediate consequence of Theorem 6.11 below. Next we have the following useful result.
Proof. Let rank Δ = , let 1 , ⋯ , be a basis of Δ and define ℛ ∶= span Λ 1 , ⋯ , . It is easily seen that ℛ is the smallest Λ -linear space containing Δ and dim Λ ℛ = rank Δ. The Λ -linear space ℛ has a proper basis and (by Corollary 4.5 ) dim Λ ℛ = dim ̂ . But clearlyR = 2 and the proof is complete. □ Let and ′ be the order chains and and ′ the order lists, respectively of submodules Δ and Δ ′ of Λ . We shall say that ′ is a subchain of , denoted ′ ⊂ if, for all , ′ ⊂ . Similarly we say that the list ′ is smaller than the list , denoted ′ ≦ if ′ ≦ for all integers . As an immediate consequence of the definition we have the following, Proposition. 6.9. Let Δ, Δ ′ ⊂ Λ be Ω − -submodules with order chains and ′ and order lists and ′ , respectively. If Δ ′ ⊂ Δ then ′ ⊂ and ′ ≦ .
Let Δ ⊂ Λ be a finitely generated Ω − -submodule. A set of elements 1 , ⋯ , ∈ Δ is called properly free if the elements are properly independent as elements of Λ (regarded as a Λ -linear space), that is, if the leading coefficientŝ 1 , ⋯ ,̂ are -linearly independent. It is then clear that if 1 , ⋯ , are properly free they are also free (i.e. independent over the ring Ω − ).
Proof. (i) We shall construct an ordered proper basis for Δ which, in particular, satisfies (6.12) and (6.13). Consider the sequence of quotient modules defined by (6.6), of which Δ is the first nonzero one. Choose any equivalence class 0 ≠ 1 Δ and let 1 ∈ Δ be any representative of 1 . Then ord 1 = Δ and 1 is clearly properly free. We proceed stepwise and assume that for > 0, 1 , ⋯ , are properly free elements of Δ satisfying (6.12) and (6.13). If < , let denote the least integer such that < . Then̂ 1 , ⋯ ,̂ ∈ are -linearly independent, but they do not span , since dim = . Thus, there exists an element +1 ∈ such that for any representative +1 ∈ +1 , the set̂ 1 , ⋯ ,̂ ,̂ +1 ∈ are -linearly independent and hence the set 1 , ⋯ , +1 is properly free. Clearly (6.13) is satisfied, and since ord +1 = so is also (6.12). By Lemma 6.8, dim Δ = rank Δ = , so that we finally obtain an ordered, properly free set of elements 1 , ⋯ , ∈ Δ satisfying (6.12) and (6.13). Let Δ ′ denote the Ω − -submodule of Λ generated by 1 , ⋯ , . It remains to be shown that Δ ′ = Δ. Obviously Δ ′ ⊂ Δ and since ord ≦ Δ for all = 1, ⋯ , and since span ̂ 1 , ⋯ ,̂ = Δ , it follows also that Δ Δ ⊂ Δ ′ . Let ∈ Δ be any element and let ord = . Then̂ ∈ whence there are elements 1 , ⋯ , > . Proceeding stepwise the same way, we conclude that there are elements ∈ Δ ′ , and since ′ ∈ Δ 2 ⊂ Δ ′ , it follows also that ∈ Δ ′ and the proof of (i) is complete. To see that (ii) holds, it suffices to observe that for each integer , every ordered proper basis 1 , ⋯ , of Δ has precisely elements whose order is less than or equal to and span ̂ 1 , ⋯ ,̂ = . □ The following immediate corollary to Theorem 6.11 gives a sharp insight to the relation between ordered proper bases of Ω − -modules and their order chain Corollary. 6.14. Let Δ ⊂ Λ be an Ω − -submodule of rank with order chain and order list . Then 1 , ⋯ , is an ordered proper basis of Δ if and only if for each ,̂ 1 , ⋯ ,̂ is a basis for .
Indeed, if ∈ ker −̄ , then̄ ( ) ∈ Ω − and there are elements 1 , ⋯ , so that ord ≧ . □ In view of Proposition 6.16, it follows that the latency kernel of a given linear i/o map̄ is finitely generated if and only if̄ is injective, the case which receives, of course, most of our attention. Before proceeding further, a remark on the noninjective case is in order Remark. 6.17. It is readily noted that if̄ ∶ Λ → Λ is a Λ -linear map, then ker −̄ can (always) be written as where ℛ is a finitely generated full Ω − -submodule of Λ . However, in the above representation, ℛ is nonunique except in the special case when̄ is injective and ker̄ = 0. If̄ 1 and̄ 2 are two Λ -linear maps then ker −̄ 1 ⊂ ker −̄ 2 if and only if ker̄ 1 + ℛ 1 ⊂ ker̄ 2 + ℛ 2 . While this condition necessarily implies ker̄ 1 ⊂ ker̄ 2 , it cannot be claimed, except in the injective case, that ℛ 1 ⊂ ℛ 2 . Hence, for computational purposes it is convenient in the noninjective case to resort to the fact that ker −̄ 1 ⊂ ker −̄ 2 if and only if Δ ̄ 1 ⊂ Δ ̄ 2 for all , where Δ ̄ is as defined in Lemma 6.15 However, Δ ̄ 1 ⊂ Δ ̄ 2 for all if and only if Δ ̄ 1 ⊂ Δ ̄ 2 for any ≦ min ord ℛ 1 , ord ℛ 2 where ℛ , = 1, 2, are any submodules in the corresponding representations of ker −̄ . By Lemma 6.15 both Δ ̄ 1 and Δ ̄ 2 are full finitely generated Ω − -submodules of Λ so that the situation is thus similar to that in the injective case. □ Let̄ ∶ Λ → Λ be an injective extended linear i∕o map and let Δ = ker −̄ . Then Δ = Ω − is a full, finitely generated Ω − -submodule of Λ and the columns 1 , ⋯ , of the generating matrix form a basis of Δ. We shall next establish certain properties of possible selections of the matrix . Proposition. 6.18. Let̄ ∶ Λ → Λ be an injective extended linear i/o map. Write ker −̄ = Ω − . Then −1 exists and is strictly causal; i.e., the elements of −1 are in −1 Ω − .
Proof. The existence of −1 follows immediately from Proposition 6.16. From the strict causality of̄ it follows that Ω − ⊂ ker −̄ , whence by Theorem 6.2 there exists a causal matrix such that = . Thus −1 = −1 and −1 is clearly strictly causal. □ Let Δ ⊂ Λ be a full finitely generated Ω − -submodule and write Δ = Ω − . We call the columns 1 , ⋯ , of a polynomial be : is of Δ if the matrix is a polynomial matrix, i.e., with elements in Ω + . We call the basis a strictly polynomial basis if its elements are strict polynomials, i.e., with elements in Ω + . If in addition is a proper basis we call it a proper polynomial basis, respectively, proper strictly polynomial basis for Δ.
Theorem. 6.21. Let̄ 1 ,̄ 2 ∶ Λ → Λ be two extended linear i/o maps with and finite dimensional K-linear spaces. There exist bicausal Λ -linear compensators̄ pr ∶ Λ → Λ and̄ po ∶ Λ → Λ such that̄ 2 =̄ po ⋅̄ 1 ⋅̄ pr if and only if there exists an order preserving Ω − -isomorphism pr : ker −̄ 2 → ker −̄ 1 . We now restrict Theorem 6.21 to the injective case to obtain the following invariance characterization of the latency indices Corollary. 6.22. Let̄ 1 ,̄ 2 ∶ Λ → Λ be two injective extended linear i/o maps with and finite dimensional -linear spaces. There exist bicausal Λ -linear compensators̄ pr∶Λ → Λ and̄ po ∶ Λ → Λ such that Proof. By the injectivity of̄ 1 and̄ 2 , both Δ 1 = ker −̄ 1 and Δ 2 = ker −̄ 2 are of rank , where = dim , and in view of Theorem 6.21 it needs only to be shown that Δ 1 and Δ 2 have the same latency indices (or latency lists) if and only if there exists an order preserving Ω − -isomorphism pr ∶ Δ 2 → Δ 1 . Let 11 , ⋯ , 1 and 21 , ⋯ , 2 be ordered proper bases for Δ 1 and Δ 2 , respectively, and let 1 and 2 be the corresponding matrices. Then an order preserving isomorphism pr ∶ Δ 2 → Δ 1 exists if and only if the matrix 1 −1 2 is bicausal which is easily seen to be the case if and only if ord 1 = ord 2 for all = 1, ⋯ , . Employing Corollary 6.14 completes the proof. □ Theorem 6.21 and Corollary 6.22 could, of course, have been stated for any Λ -linear maps and not only strictly causal ones. The proofs did in no way depend on the causality properties of the maps involved. Also, Corollary 6.22 could have been obtained as an application of the existence of, so called, Smith canonical forms for matrices over Euclidean rings (see, e.g., MacDuffee [1934]).
In general, the precompensator remainder̄ is dynamic and can be represented as̄ = +̄ where is the static part of̄ and̄ ∶ Λ → Λ is strictly causal, i.e., an extended linear i/o map. We recall (see, in particular, Hautus and Heymann [1978]) that the dynamic characteristics of̄ are determined by ker +̄ ⋅ + which is an Ω + -submodule of Ω + and can be represented by where is a polynomial matrix whose columns form a basis for ker +̄ ⋅ + . The degree of the determinant of (when is nonsingular) is the dimension of the minimal state space realizinḡ . More specifically, if in (7.1) is selected to be proper, i.e., the columns of are properly free (in the sense that the leading coefficient vectors are -linearly independent just as in 84 above), then the column degrees , = 1, ⋯ , are the reachability indices of̄ and their sum is ∑ =1 = = deg ⋅ det . It is of interest in selecting a (̄ ,̄ ) pair representing a given precompensator̄ to choose the representation in such a way that the precompensator remainder̄ has least dynamic order, i.e., is realizable by a state space of least possible dimension. In this way the precompensator is realized "as much as possible" by feedback. The following theorem provides a bound on the dynamic order of the precompensator remainder̄ which need not be exceeded in the realization of any bicausal precompensator̄ , and which is dependent only on the dynamic properties (latency) of the i/o map̄ under consideration. Proof. By Hautus and Heymann [1978, Theorem 6.11] the lemma will be proved upon showing that there exists an order-preserving Ω + -isomorphism ker +̄ ⋅ + → ker +̄ −1 ⋅ + . We shall see that the map̄ itself, which is in particular also an order preserving Ω + -isomorphism, satisfies the required properties. Indeed, let ∈ ket +̄ ⋅ + be any element. Then̄ ⋅ + =̄ ∈ Ω + and since also ∈ Ω + we have =̄ −1 (̄ ) =̄ −1 + (̄ ) ∈ Ω + , whencē ∈ ker +̄ −1 ⋅ + , completing the proof. □ Let̄ ∶ Λ → Λ be an injective extended linear i/o map and let 1 , ⋯ , be a proper strictly polynomial basis for ker −̄ (see Theorem 6.19 ), and write ker −̄ = Ω − where = 1 , ⋯ , . Then −1 is also polynomial and the column degrees of −1 are (by definition) the latency indices of̄ . Below we shall not distinguish sharply between maps and their transfer functions. Let − ∶ Λ → Ω − ∶ Σ − ↦ Σ ≥0 − denote the causal truncation. Let ∶ Λ → Ω − be defined as the (unique) Λ -linear map whose transfer function is given by