Natural Factorization of Linear Control Systems through Parallel Gathering of Simple Systems

Carriegos, M. V.

doi:https://doi.org/10.1155/2023/7963973

Journal of Mathematics

On this page

Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 7963973 | https://doi.org/10.1155/2023/7963973

Natural Factorization of Linear Control Systems through Parallel Gathering of Simple Systems

M. V. Carriegos¹

Academic Editor: Firdous A. Shah

Received12 Aug 2022

Revised11 Apr 2023

Accepted18 Apr 2023

Published20 May 2023

Abstract

Linear systems over vector spaces and feedback morphisms form an additive category taking into account the parallel gathering of linear systems. This additive category has a minimal exact structure and thus a notion of simple systems as those systems have no subsystems apart from zero and themselves. The so-called single-input systems are proven to be exactly the simple systems in the category of reachable systems over vector spaces. The category is also proven to be semisimple in objects because every reachable linear system is decomposed in a finite parallel gathering of simple systems. Hence, decomposition result is fulfilled for linear systems and feedback morphisms, but category of reachable linear systems is not abelian semisimple because it is not balanced and hence fails to be abelian. Finally, it is conjectured that the category of linear systems and feedback actions is in fact semiabelian; some threads to find the result and consequences are also given.

1. Introduction

Mathematical study of control systems arises from engineering after seminal work of Maxwell [1] on flyball governors of steam engines. Linear systems are found almost everywhere in control theory [2] both as linear models or as linearizations. In particular, the algebraic study of linear systems in the state-space approach [3] deals with linear systems defined on algebras and modules over a commutative ring [4, 5]. This approach has been used recently in the field of convolutional codes [6–10]. Convolutional codes are in fact error-correcting codes over a finite field defined as vector subspaces of , where is the field of rationals which are realized as linear control systems over .

Feedback is the main tool in the state-space approach [11]: two linear systems are feedback equivalent if one can be transformed in the other by means of a feedback action. The feedback equivalence relation has been studied for decades from many different perspectives: Kronecker invariants associated to a linear system were found in [12] to characterize when two reachable linear systems (scalars in a field ) are feedback equivalent; differential geometric tools were applied in [13] to study feedback equivalence of holomorphic pairs of matrices; linear systems over finitely generated -modules were suggested as algebraic tool to study parametric systems and integer valued systems [4, 5]; and later -module invariants were found [14–16]. On the other hand, if , then (see [4], p. 276) a linear system gives rise to a vector bundle over the Riemann sphere and equivalence of linear systems is characterized by the isomorphism of their associated vector bundles. The reader can see Lomadze’s article [17] as one main reference in the use of vector bundle decomposition results (mainly Grothendieck’s Theorem) in order to classify linear systems over -vector spaces.

Several algebraic [18, 19] and geometric objects have been employed to study linear systems. Casti [4] (p. 292) quotes that each gadget, whether it be a -module, a vector bundle, or a Grassmann variety, illuminates a different aspect of the overall category called linear systems. However, Casti defined the category of realizations of behaviours [4], p. 304, but not the category of linear systems itself. Another topic was studied using categories: Brewer and Klingler [20] proved that if is a commutative ring containing a nonzero finitely generated maximal ideal containing its annihilator, such that every unit of lifts to a unit of , then the category of reachable systems over is “wild” in the sense of classical representation theory. From this it follows that a canonical form for a reachable system over a principal ideal domain is not likely to be found. More specifically, canonical forms are unlikely to be found for systems over and . Precisely, the impossibility of finding out canonical forms for arbitrary linear systems over rings is one of the main motivations of introducing categories of linear systems as a way to detect feedback invariants of linear systems.

Category of linear systems and feedback morphisms was defined in [21] as the category whose objects are linear systems over a fixed commutative ring and whose morphisms are the feedback morphisms. A feedback morphism is a linear map between state spaces preserving the dynamics up to feedback actions. From this point of view, feedback actions are exactly the isomorphisms in the category (no matter the scalar ring ). Moreover a collection of feedback invariants is found in [12]. This set of invariants generalizes both Kronecker’s invariants over fields and -module invariants found in [14] to a complete set of feedback invariants, not in the case of arbitrary linear systems but in the case of regular systems [12]. Hence, an answer to Brewer and Klingler’s negative result would be as follows: though trying to classify all reachable linear systems is a wild problem, the classification of all regular linear systems is given over any commutative ring [21]. Moreover, both feedback equivalence and so-called dynamic feedback equivalence [22, 23] can be studied by taking into account the -theory groups of the symmetric monoidal category of regular systems [21].

The notion of feedback morphism was introduced in [21] to circumscribe the problem of feedback classification of linear systems over -modules as a linear map preserving both the dynamics and controls of systems (see [24] as an early article in the use of dynamorphisms). But the notion of feedback morphism is interesting by itself in terms of linear systems over -vector spaces [25]. Kernels and cokernels of feedback morphisms of linear systems were introduced in [25] and showed that the category has all cokernels. It was conjectured that the category has all kernels as well, and hence will be preabelian. But despite this conjecture, it is proved that is not abelian because it is not balanced (i.e., there are bimorphisms that are not isomorphisms).

This article focuses exact structures [26, 27] onto the additive category of linear systems over vector spaces and parallel gathering (biproduct) of linear systems. The goal of this article is to find out the simple objects in that category. The results are that the simple objects, for the minimal exact structure [27], are exactly the classical canonical controller forms [2] adapted to this framework. On the other hand, the category is semisimple in objects because Brunovsky’s theorem [12] states, in our framework, that every reachable linear system is parallel gathering (biproduct) of a finite number of simple systems. Hence, a solution of decomposition problems [24] is given in terms of feedback actions instead only for dynamorphisms. Finally, note that the category itself fails to be semisimple because is not abelian and Schur’s lemma does not hold in this category [28].

The article is structured as follows. Main definitions: linear system, feedback morphism, reachable system, parallel gathering, and decomposition are found in the second section “categories of linear systems”. Then, third section “simple systems over vector spaces” is devoted to give the main results, the characterization of simple systems over a vector space, and that every reachable system decomposes as finite parallel gathering of simple systems. Minimal exact structure is introduced as well. Canonical controller forms and Brunovsky’s theorem are stated in our categorical framework. Finally, we give some concluding remarks, where some results are highlighted and some threads to develop our results are given.

2. Categories of Linear Systems

A linear system is a triple where is a finite-dimensional -vector space, is a linear map, and is a vector subspace (we will use to denote vector subspace). The category of -vector spaces and linear maps is denoted by . The category of linear systems over finite dimensional -vector spaces gathers linear systems as objects in the category and feedback morphisms as morphisms in the category.

Definition 1. (see [21], Defintion 3.2.) A feedback morphism is given by a linear map satisfying the following properties:(i)(ii).The pair is a category. In fact, it is a -linear category (i.e., enriched on the category of -vector spaces and linear maps). The functor forget-the-dynamics given by in objects and by in morphisms is obviously injective on morphisms, hence is a faithful functor. is also a dense functor because every vector space occurs as . But functor is not full because not every linear map arises as a feedback morphism, i.e., the induced map is injective, but it is not surjective in general. A linear combination of feedback morphisms is a feedback morphism hence the set of feedback morphisms between two linear systems is a vector subspace of linear maps between state spaces. Feedback morphisms generalize the notion of feedback equivalence ([21], Proposition 3.3.). In fact, two linear systems and are feedback equivalent (in the classical sense) exactly when they are isomorphic in , i.e., when there exists a feedback morphism such that its inverse is also a feedback morphism. Note that the inverse of a feedback morphism is not a feedback morphism in general even in the case of underlying linear map happens to be invertible: the morphism given by does not admit an inverse as feedback morphism [25], and hence is not an isomorphism in , but however is both monic and epic in the category, and thus a bimorphism in , while is a isomorphism in .

2.1. Linear System Decomposition

Categories of linear systems are additive. The parallel gathering of systems and given by the following equation:is a biproduct (both product and coproduct) in ([21], Lemma 3.5.). Hence, the category is additive, in fact symmetric monoidal. The zero object is linear system .

A linear system is indecomposable if whenever one has , then one has that either or . Next section is devoted to prove the main result of this article which is the canonical decomposition of reachable linear systems over vector spaces. First of all, we recall the definition and some key properties of reachable systems.

Consider a linear system and the subspaces of given recursively by , and in general . This sequence of subspaces is an ascending chain (i.e., ). The chain is strict up to an index (the degree of the linear system) , and from this index, the chain stabilizes forever

Since is finite dimensional, it follows that . We will often use the notation to denote . Linear system is called reachable if . Note that the zero system is reachable of degree 0.

Denote by the full subcategory of collecting all reachable linear systems and all feedback morphisms in between them. If linear systems are reachable then parallel gathering system is also reachable. Hence, is internal to and reachable systems is also an additive category, in fact it is also symmetric monoidal. Consider the restriction of forget-the-dynamics functor to reachable systems . Functor is newly injective on morphisms, hence is faithful. Functor is also dense because every vector space occurs as , and is trivially a reachable linear system. But is not full because the induced liner map between -vector spaces is not surjective in general.

3. Simple Systems over Vector Spaces

This section contains the main result of this article (Theorem 5), which states that simple reachable linear systems over a vector space are exactly those linear systems , where . This result will be proven by using exact structures on the category of reachable linear systems. Finally, simple systems will arise as those nonzero reachable linear systems have no strict subsystems.

3.1. Exact Structure on the Category of Linear Systems

Bühler’s systematic elementary expository article [26] is followed in the sequel in order to recall exact categories from additive ones and to state the minimal exact structure on additive category . The exact structure , though minimal, will be enough to find out simple linear systems.

Definition 2. (see [26], Definition 2.1.) A kernel-cokernel pair in is a pair of composable feedback morphisms such that is the kernel of , and is the cokernel of . This fact is denoted by . If a class of kernel-cokernel pairs is fixed, an admissible monic is a morphism such that there exists a morphism such that . An admissible epic is defined dually. An exact structure is a class of kernel-cokernel pairs which is closed under isomorphisms of linear systems and satisfies the following axioms:(i)The identity of the zero object is an admissible epic.(ii)The class of admissible monics and the class of admissible epics are closed under composition.(iii)The push-out of admissible monic along feedback map yields an admissible monic.(iv)The pull-back of admissible epic along feedback map yields an admissible epic. There would be several exact structures on . The next result remarks that we have at least a minimal exact structure which gathers all kernel-cokernel feedback pairs isomorphic to a splitting pair.

Theorem 3. The kernel-cokernel pairs isomorphic to form a exact structure , and every other exact structure contains .

Proof. (see [27], Proposition 2.12).

3.2. Simple Linear Systems

In the sequel, we consider the exact structure and the exact category . Next, we define the admissible subsystem in terms of the concept of admissible subobject in an exact category.

Definition 4. [27], 3.1.). System is an admissible subsystem of (denoted by ) if there exists an admissible section or equivalently if one has for some in . A nonzero system is -simple if every subsystem verifies either or .
Now, we are ready to deal with the main result of the article.

Theorem 5. Simple reachable systems are exactly those linear systems such that

Proof. The proof involves the categorical version of some classical results which we will prove later (Lemmas 7 and 9) together with a dimension computation result Lemma 8. The proof of Theorem 5, up to these results, works as follows:
Set a field and consider the exact category of reachable linear systems . Consider a reachable linear system , then by Lemma 9, we have that where , and systems are not zero, and . It follows that is an admissible section, hence is a nonzero subsystem of . Therefore, in order to being simple, it is necessary that , , and .
Conversely, consider a reachable subsystem where and let us prove that is simple. The proof is performed by contradiction. Assume that and that is not simple. Then, there exists a nonzero subsystem , that is to say, there exists an admissible section . Newly by Lemma 9, we can suppose that , where , , and . Put , then is also an admissible section, and therefore, we may assume without loss that . On the other hand, verifies itself that and hence, by Lemma 7, one has .
Therefore, the situation we have reached is that is an admissible section. Since , it follows by Lemma 8 that . On the other hand, every functor preserves sections, then is a section and therefore . Consequently, and newly by Lemma 8, because of , one has that and that is simple because every nonzero subsystem of must be isomorphic to
Now, the next paragraphs are devoted to prove the results we used above.

Definition 6. (canonical controller form). A linear system , where is called a single-input system. The linear system , where is the Jordan block of size and eigenvalue 0, and is the subspace spanned by the first vector of standard basis of will be called canonical controller form of size . In other words,A classical result in control theory shows that feedback classification is trivial for single-input reachable systems. Next, we state this result in our categorical framework:

Lemma 7. Let be a reachable system and , then

Proof. Let . Since is reachable, it follows that , and hence is a basis of . Let be the characteristic polynomial of . Now, consider the basis of given by the following equation:Then, linear system is written in above basis as follows:which is feedback isomorphic to by means of identity morphism because .
Morphisms between single-input reachable systems are studied in [29]. The dimension of the vector space of feedback morphisms between two single-input reachable systems is computed as follows:

Lemma 8 (see [29], 6.3.). The dimension of the space of feedback morphisms between canonical controller forms is as follows:

Proof. Let be any feedback morphism. Consider the matrix of in the standard bases and . Since is a feedback morphism, it follows:(1) and therefore (2) and therefore(i) for all and all (ii)Now consider the abovementioned restrictions in the three cases:
: from (i), we have that is on the formNow, from (i) and (ii), we have that matrixdepends exactly on free parameters .
In these case restrictions, (i) and (ii) yield that is on the form. Finally in this case, the restrictions yield that the only possibility is .
The standard decomposition result on reachable linear systems is the Brunovsky’s theorem [12]. The result in our categorical setting is as follows.

Lemma 9. (Brunovsky’s theorem). Let be a reachable linear system (in ) and set . Then, there exists a partition of integer such that

Proof. By recursion in , consider a nonzero vector and linear system . Natural inclusion gives raise to an exact sequence in Now, . System is reachable, and note that . Hence, by recursion, where all are in .
Because , it follows by Lemma 7 that every system is isomorphic to a system on the form . This concludes the proof.

4. Conclusion

The category of linear systems over vector spaces and feedback actions is studied. The parallel gathering of linear systems is biproduct and thus category is additive and has the minimal natural exact structure given by split exact sequences. Single-input systems are shown to be the simple objects in , and on the other hand, every object is a parallel gathering of simple objects. Thus, the following result is obtained.

Corollary 10. Category is object semisimple, i.e., every nonzero object is isomorphic to a finite biproduct of simple objects.

Because category is not balanced, it follows that it is not abelian and a fortiori and it is not abelian semisimple, that is to say, though simple systems are bricks in the sense of Enomoto’s article ([28], Definition 2.1), the second statement of Schur’s lemma does not hold in because morphisms between nonisomorphic bricks are not zero. Note for instance that , thus we do not have a matrix-like representation for morphisms of linear systems, at least in the sense of Schur’s lemma.

To conclude, we would like to point out some lines of further work. It is known that categories of linear systems have cokernels ([29], Theorem 3.4.) and that it is conjectured that categories of linear systems have kernels as well. First task is to compute effectively kernels of feedback morphisms. Once this is fulfilled, the category would be proven to be preabelian ([27], Definition 2.5.), and because every feedback morphism would have kernel and cokernel, it follows that image and coimage of every feedback morphism are obtained.

The second task is to check that canonical morphism from the coimage to the image of a feedback morphism is always a bimorphism. We conjecture that this is true and hence categories of linear systems are semiabelian ([27], Definition 2.5.) ([26], 4.10).

Even more, we conjectured that kernels and cokernels are stable ([26], Definition 4.1.). Hence, because kernels and cokernels are stable in , it would follow by Schneiders’ result ([26], Proposition 4.4.) ([30], 1.1.7.) that the class of all kernel-cokernel pairs is the maximal and thus the natural, exact structure on . The third task is to prove the former and to show that our decomposition results are also the decomposition results taking the natural exact structure.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares no conflicts of interest.

Acknowledgments

The author would like to thank the CAFE (Ciberseguridad, Aplicaciones, Fundamentos y Educación) research group for its support.

References

J. C. Maxwell, “On governors,” Proceedings of the Royal Society of London, vol. 16, pp. 270–283, 1867.
View at: Google Scholar
K. Ogata, Modern Control Engineering, Prentice Hall, Hoboken, NJ, USA, 2010.
E. D. Sontag, Mathematical Control Theory, Springer, Berlin, Germany, 1991.
J. L. Casti, Linear Dynamical Systems, Academic Press, Cambridge, MA, USA, 1987.
J. W. Brewer, J. W. Bunce, and F. S. VanVleck, Linear Dynamical Systems over Commutative Rings, Marcel Dekker, New York, NY, USA, 1986.
N. DeCastro-García, “Feedback equivalence of convolutional codes over finite rings,” Open Mathematics, vol. 15, no. 1, pp. 1495–1508, 2017.
View at: Publisher Site | Google Scholar
A. L. Muñoz Castañeda, N. DeCastro-García, and M. V. Carriegos, “On the state approach representations of convolutional codes over rings of modular integers,” Mathematics, vol. 9, no. 22, p. 2962, 2021.
View at: Publisher Site | Google Scholar
J. M. Muñoz Porras and J. I. Iglesias Curto, “Classification of convolutional codes,” Linear Algebra and Its Applications, vol. 432, no. 10, pp. 2701–2725, 2010.
View at: Publisher Site | Google Scholar
V. Herranz, D. Napp, and C. Perea, “Serial concatenation of a block code and a 2d convolutional code,” Multidimensional Systems and Signal Processing, vol. 30, no. 3, pp. 1113–1127, 2019.
View at: Publisher Site | Google Scholar
D. Napp, R. Pinto, and C. Rocha, “State representations of convolutional codes over a finite ring,” Linear Algebra and Its Applications, vol. 640, pp. 48–66, 2022.
View at: Publisher Site | Google Scholar
R. E. Kalman, “Kronecker invariants and feedback,” in Ordinary Differential Equations, Academic Press, Cambridge, MA, USA, 1972.
View at: Google Scholar
P. A. Brunovsky, “A classification of linear controllable systems,” Kibernetika, vol. 3, 1970.
View at: Google Scholar
J. Ferrer, M. García, and F. Puerta, “Brunowsky local form of a holomorphic family of pairs of matrices,” Linear Algebra and Its Applications, vol. 253, no. 1–3, pp. 175–198, 1997.
View at: Publisher Site | Google Scholar
J. A. Hermida-Alonso, M. P. Pérez, and T. Sánchez-Giralda, “Brunovsky's canonical form for linear dynamical systems over commutative rings,” Linear Algebra and Its Applications, vol. 233, pp. 131–147, 1996.
View at: Publisher Site | Google Scholar
M. V. Carriegos, “Enumeration of classes of linear systems via equations and via partitions in an ordered abelian monoid,” Linear Algebra and Its Applications, vol. 438, no. 3, pp. 1132–1148, 2013.
View at: Publisher Site | Google Scholar
M. V. Carriegos and N. DeCastro-García, “Partitions of elements in a monoid and its applications to systems theory,” Linear Algebra and Its Applications, vol. 491, pp. 161–170, 2016.
View at: Publisher Site | Google Scholar
V. Lomadze, “Applications of vector bundles to factorization of rational matrices,” Linear Algebra and Its Applications, vol. 288, pp. 249–258, 1999.
View at: Publisher Site | Google Scholar
M. A. Arbib and H. P. Zeiger, “On the relevance of abstract algebra to control theory,” Automatica, vol. 5, pp. 589–606, 1969.
View at: Publisher Site | Google Scholar
M. A. Arbib, “Automata theory and control theory - a rapprochement,” Automatica, vol. 3, no. 3-4, pp. 161–189, 1966.
View at: Publisher Site | Google Scholar
J. W. Brewer and L. Klingler, “On feedback invariants for linear dynamical systems,” Linear Algebra and Its Applications, vol. 325, no. 1–3, pp. 209–220, 2001.
View at: Publisher Site | Google Scholar
M. V. Carriegos and A. L. Muñoz Castañeda, “On the K-theory of feedback actions on linear systemsK-theory of feedback actions on linear systems,” Linear Algebra and Its Applications, vol. 440, pp. 233–242, 2014.
View at: Publisher Site | Google Scholar
J. W. Brewer and L. Klingler, “Dynamic feedback over commutative rings,” Linear Algebra and Its Applications, vol. 98, pp. 137–168, 1988.
View at: Publisher Site | Google Scholar
J. A. Hermida-Alonso and M. T. Trobajo, “The dynamic feedback equivalence over principal ideal domains,” Linear Algebra and Its Applications, vol. 368, pp. 197–208, 2003.
View at: Publisher Site | Google Scholar
M. A. Arbib and E. G. Manes, “Foundations of system theory: decomposable systems,” Automatica, vol. 10, no. 3, pp. 285–302, 1974.
View at: Publisher Site | Google Scholar
M. V. Carriegos, “Some categorical properties of linear systems,” Mathematics, vol. 10, no. 12, p. 2088, 2022.
View at: Publisher Site | Google Scholar
T. Bühler, “Exact categories,” Expositiones Mathematicae, vol. 28, no. 1, pp. 1–69, 2010.
View at: Publisher Site | Google Scholar
T. Brüstle, S. Hassoun, D. Langford, and S. Roy, “Reduction of exact structures,” 2021, https://arxiv.org/abs/1809.01282.
View at: Google Scholar
H. Enomoto, “Schur’s lemma for exact categories implies abelian,” Journal of Algebra, vol. 584, pp. 260–269, 2021.
View at: Publisher Site | Google Scholar
M. V. Carriegos, Morphisms of Linear Control Systems, Open Math, Colorado, CO, USA, 2022.
J. P. Schneiders, “Quasi-abelian categories and sheaves,” Mémoires de la Société Mathématique de France, vol. 1, pp. 1–140, 1999.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2023 M. V. Carriegos. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

160

Downloads

231

Citations