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Recent progress in an algebraic analysis approach to linear systems

Multidimensional Systems and Signal Processing volume 26, pages 349–388 (2015)Cite this article

Abstract

This paper addresses systems of linear functional equations from an algebraic point of view. We give an introduction to and an overview of recent work by a small group of people including the author of this article on effective methods which determine structural properties of such systems. We focus on parametrizability of the behavior, i.e., the set of solutions in an appropriate signal space, which is equivalent to controllability in many control-theoretic situations. Flatness of the linear system corresponds to the existence of an injective parametrization. Using an algebraic analysis approach, we associate with a linear system a module over a ring of operators. For systems of linear partial differential equations we choose a ring of differential operators, for multidimensional discrete linear systems a ring of shift operators, for linear differential time-delay systems a combination of those, etc. Rings of these kinds are Ore algebras, which admit Janet basis or Gröbner basis computations. Module theory and homological algebra can then be applied effectively to study a linear system via its system module, the interpretation depending on the duality between equations and solutions. In particular, the problem of computing bases of finitely generated free modules (i.e., of computing flat outputs for linear systems) is addressed for different kinds of algebras of operators, e.g., the Weyl algebras. Some work on computer algebra packages, which have been developed in this context, is summarized.

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Notes

  1. The assumption that \(D\) is a left Noetherian domain actually implies (cf. McConnell and Robson 2000, Thm. 2.1.15) that \(D\) satisfies the left Ore condition, i.e., every pair of non-zero elements of \(D\) has a non-zero common left multiple, a property shared by all Ore algebras (cf. also Proposition 3.7).

  2. As mentioned earlier, every pair of non-zero elements of \(D\) has a non-zero common left multiple, which implies, e.g., that \({{\mathrm{t}}}(M)\) is a left \(D\)-module.

  3. We thank Burt Totaro for a clarification of this statement and the reference Cohn (2006).

  4. In concrete examples we may assume that the matrix \(R\) is defined over a computable subalgebra of \(D\).

  5. For Janet bases the divisibility relation of terms is actually a restriction of the usual divisibility relation. The concept of Janet division (or, more generally, of an involutive division) determines for each monomial the set of indeterminates which may be multiplied from the left to the monomial when it is used for reduction of other terms. As a consequence, every element of \(D^{1 \times q} \, R\) has a unique representation as left \(D\)-linear combination of the Janet basis elements taking their so-called multiplicative variables into account. For a survey on the algorithmic development of this efficient alternative to Buchberger’s algorithm we refer to Gerdt (2005).

References

  • Avanessoff, D., & Pomet, J.-B. (2007). Flatness and Monge parameterization of two-input systems, control-affine with 4 states or general with 3 states. ESAIM: Control, Optimisation and Calculus of Variations, 13(2), 237–264 (electronic).

    Google Scholar 

  • Barakat, M. (2001). Jets. A MAPLE-package for formal differential geometry. In V. G. Ganzha, E. W. Mayr, & E. V. Vorozhtsov (Eds.), Computer algebra in scientific computing CASC 2001 (pp. 1–12). Berlin: Springer.

  • Barakat, M. (2010). Purity filtration and the fine structure of autonomy. In Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2010), Budapest, Hungary, 2010. Also in E. Zerz (Ed.) Algebraic Systems Theory, Behaviors, and Codes (pp. 1–6). Aachen: Shaker-Verlag.

  • Barakat, M., & Robertz, D. (2008). homalg: A meta-package for homological algebra. Journal of Algebra and its Applications, 7(3), 299–317.

    Article  MATH  MathSciNet  Google Scholar 

  • Björk, J.-E. (1979). Rings of differential operators. Volume 21 of North-Holland Mathematical Library. Amsterdam: North-Holland Publishing Co.

  • Blinkov, Y.A., Cid, C.F., Gerdt, V.P., Plesken, W., & Robertz, D. (2003) The MAPLE Package “Janet”: I. Polynomial Systems. II. Linear Partial Differential Equations. In V.G. Ganzha, E.W. Mayr, E.V. Vorozhtsov (eds.) Proceedings of the 6th International Workshop on Computer Algebra in Scientific Computing (pp. 31–40 resp. 41–54). Passau (Germany), 2003. http://wwwb.math.rwth-aachen.de/Janet

  • Boudellioua, M. S., & Quadrat, A. (2010). Serre’s reduction of linear functional systems. Mathematics in Computer Science, 4(2–3), 289–312.

    Article  MATH  MathSciNet  Google Scholar 

  • Boulier, F., Lazard, D., Ollivier, F., & Petitot, M. (2009). Computing representations for radicals of finitely generated differential ideals. Applicable Algebra in Engineering, Communication and Computing, 20(1), 73–121.

    Article  MATH  MathSciNet  Google Scholar 

  • Bourbaki, N. (1980). Algèbre, Chapter 10. Algèbre homologique. Paris: Masson.

  • Bourlès, H., & Oberst, U. (2012). Elimination, fundamental principle and duality for analytic linear systems of partial differential-difference equations with constant coefficients. Mathematics of Control, Signals, and Systems, 24(4), 351–402.

    Article  MATH  MathSciNet  Google Scholar 

  • Buchberger, B. (2006). An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal. Journal of Symbolic Computations, 41(3–4), 475–511. Translated from the 1965 German original by Michael P. Abramson.

    Google Scholar 

  • Caro, N., & Levcovitz, D. (2010). On a Theorem of Stafford. Cadernos de Mathemática, 11, 63–70.

    Google Scholar 

  • Chakhar, A., Cluzeau, T., & Quadrat, A. (2011). An algebraic analysis approach to certain classes of nonlinear partial differential systems. In Proceedings of the 7th international workshop on multidimensional (nD) Systems. France: Poitiers.

  • Chyzak, F. (1998). Fonctions holonomes en calcul formel. Palaiseau, France: Ph.D. thesis, Ecole Polytechnique.

  • Chyzak, F., Quadrat, A., & Robertz, D. (2005). Effective algorithms for parametrizing linear control systems over Ore algebras. Applicable Algebra in Engineering, Communication and Computing, 16(5), 319–376.

    Article  MATH  MathSciNet  Google Scholar 

  • Chyzak F., Quadrat A., & Robertz D., (2007). OreModules: a symbolic package for the study of multidimensional linear systems. In J. Chiasson, J.-J. Loiseau (Eds.) Applications of time delay systems. Volume 352 of Lecture Notes in Control and Information Sciences (pp. 233–264). Berlin: Springer http://wwwb.math.rwth-aachen.de/OreModules

  • Chyzak, F., & Salvy, B. (1998). Non-commutative elimination in Ore algebras proves multivariate identities. Journal of Symbolic Computation, 26, 187–227.

    Article  MATH  MathSciNet  Google Scholar 

  • Cluzeau, T., & Quadrat, A. (2008). Factoring and decomposing a class of linear functional systems. Linear Algebra and Its Applications, 428(1), 324–381.

    Article  MATH  MathSciNet  Google Scholar 

  • Cluzeau, T., & Quadrat, A. (2009). OreMorphisms: a homological algebraic package for factoring, reducing and decomposing linear functional systems. In J.-J. Loiseau, W. Michiels, S.-I. Niculescu, R. Sipahi (eds.), Topics in time delay systems. Volume 388 of Lecture Notes in Control and Information Sciences (pp. 179–194). Springer, Berlin. Cf. also http://www-sop.inria.fr/members/Alban.Quadrat/OreMorphisms or http://perso.ensil.unilim.fr/cluzeau/OreMorphisms

  • Cluzeau, T., & Quadrat, A. (2012). Serre’s reduction of linear partial differential systems with holonomic adjoints. Journal of Symbolic Computation, 47(10), 1192–1213.

    Article  MATH  MathSciNet  Google Scholar 

  • Cohn, P. M. (2006). Free ideal rings and localization in general rings. Volume 3 of New Mathematical Monographs. Cambridge: Cambridge University Press.

  • Coutinho, S. C., & Holland, M. P. (1988). Module structure of rings of differential operators. Proceedings of the London Mathematical Society, 57, 417–432.

    Article  MATH  MathSciNet  Google Scholar 

  • Diop, S. (1992). Differential-algebraic decision methods and some applications to system theory. Theoretical Computer Science, 98, 137–161. Second Workshop on Algebraic and Computer-theoretic Aspects of Formal Power Series (Paris, 1990).

  • Ehrenpreis, L. (1970). Fourier analysis in several complex variables. Volume XVII of Pure and Applied Mathematics. New York-London-Sydney: Wiley-Interscience Publishers, Wiley.

  • Eisenbud, D. (1995). Commutative algebra. Volume 150 of Graduate Texts in Mathematics. New York: Springer. With a view toward algebraic geometry.

  • Fabiańska, A. (2009). Algorithmic analysis of presentations of groups and modules. Germany: Ph.D. thesis, RWTH Aachen University. This thesis is available at http://darwin.bth.rwth-aachen.de/opus3/volltexte/2009/2950. QuillenSuslin project (http://wwwb.math.rwth-aachen.de/QuillenSuslin)

  • Fabiańska, A., & Quadrat, A. (2006). Flat multidimensional linear systems with constant coefficients are equivalent to controllable 1-D linear systems. In Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2006) (pp. 560–582). Kyoto: Japan.

  • Fabiańska, A., & Quadrat, A. (2007). Applications of the Quillen-Suslin theorem tomultidimensional systems theory. In H. Park & G. Regensburger (Eds.), Gröbner bases in control theory and signal processing. Volume 3 of Radon Series on Computational and Applied Mathematics (pp. 23–106). Berlin:Walter de Gruyter.

  • Fliess, M. (1991). Controllability revisited. In A. C. Antoulas (Ed.), Mathematical system theory (pp. 463–474). Berlin: Springer. The influence of R. E. Kalman.

    Chapter  Google Scholar 

  • Fliess, M., Lévine, J., Martin, P., & Rouchon, P. (1995). Flatness and defect of non-linear systems: Introductory theory and examples. International Journal of Control, 61(6), 1327–1361.

    Article  MATH  MathSciNet  Google Scholar 

  • Fliess, M., & Mounier, H. (1998). Controllability and observability of linear delay systems: An algebraic approach. ESAIM: Control. Optimisation and Calculus of Variations, 3, 301–314.

    Article  MATH  MathSciNet  Google Scholar 

  • Freund, E. (1971). Zeitvariable Mehrgrößensysteme. Volume 57 of Lecture Notes in Operations Research and Mathematical Systems. Heidelberg: Springer.

  • Fröhler, S., & Oberst, U. (1998). Continuous time-varying linear systems. Systems & Control Letters, 35(2), 97–110.

    Article  MATH  MathSciNet  Google Scholar 

  • Gago-Vargas, J. (2003). Bases for projective modules in \(A_n(k)\). Journal of Symbolic Computation, 36(6), 845–853.

    Article  MATH  MathSciNet  Google Scholar 

  • Gallego, C., & Lezama, O. (2011). Gröbner bases for ideals of \(\sigma \text{- }PBW\) extensions. Communications in Algebra, 39(1), 50–75.

    Article  MATH  MathSciNet  Google Scholar 

  • Gerdt, V. P. (2005). Involutive algorithms for computing Gröbner bases. In S. Cojocaru, G. Pfister, & V. Ufnarovski (Eds.), Computational commutative and non-commutative algebraic geometry, NATO Science Series (pp. 199–225). Amsterdam: IOS Press.

    Google Scholar 

  • Gluesing-Luerssen, H. (2002). Linear delay-differential systems with commensurate delays: An algebraic approach. Volume 1770 of Lecture Notes in Mathematics. Berlin: Springer.

  • Grigor’ev, D Yu. (1989). Complexity of quantifier elimination in the theory of ordinary differentially closed fields. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 176, 53–67.

    MATH  Google Scholar 

  • Guo, L., Regensburger, G., & Rosenkranz, M. (2014). On integro-differential algebras. Journal of Pure and Applied Algebra, 218(3), 456–473.

    Article  MATH  MathSciNet  Google Scholar 

  • Hillebrand, A., & Schmale, W. (2001). Towards an effective version of a theorem of Stafford. Journal of Symbolic Computation, 32, 699–716.

    Article  MATH  MathSciNet  Google Scholar 

  • Janet, M. (1929). Leçons sur les systèmes d’équations aux dérivées partielles. Cahiers Scientifiques, fasc. IV. Paris: Gauthiers-Villars.

  • Janet, M. (1971). P. Zervos et le problème de Monge. Bull. Sci. Math. (2), 95, 15–26.

  • Kandri-Rody, A., & Weispfenning, V. (1990). Noncommutative Gröbner bases in algebras of solvable type. Journal of Symbolic Computation, 9(1), 1–26.

    Article  MATH  MathSciNet  Google Scholar 

  • Kashiwara, M. (1995). Algebraic study of systems of partial differential equations, Volume 63. France (N.S.): Mém. Soc. Math.

  • Kredel, H. (1993). Solvable polynomial rings. Aachen: Shaker.

    MATH  Google Scholar 

  • Lam, T. Y. (1999). Lectures on modules and rings. Volume 189 of Graduate Texts in Mathematics. New York: Springer.

  • Lam, T. Y. (2006). Serre’s problem on projective modules. Springer Monographs in Mathematics. Berlin: Springer.

    Book  Google Scholar 

  • Lange-Hegermann, M., & Robertz, D. (2013). Thomas decompositions of parametric nonlinear control systems. In Proceedings of the 5th Symposium on System Structure and Control (pp. 291–296). France: Grenoble.

  • Levandovskyy, V. (2005). Non-commutative computer algebra for polynomial algebras: Gröbner bases, applications and implementation. Germany: Ph.D. thesis, University of Kaiserslautern. This thesis is available at http://kluedo.ub.uni-kl.de/volltexte/2005/1883

  • Levandovskyy, V., & Zerz, E. (2005). Computer algebraic methods for the structural analysis of linear control systems. In Proceedings in Applied Mathematics and Mechanics (PAMM). Wiley-VCH. Cf. also http://www.math.rwth-aachen.de/Eva.Zerz/CLIPS/

  • Lévine, J. (2011). On necessary and sufficient conditions for differential flatness. Applicable Algebra in Engineering, Communication and Computing, 22(1), 47–90.

    Article  MATH  MathSciNet  Google Scholar 

  • Leykin, A. (2004). Algorithmic proofs of two theorems of Stafford. Journal of Symbolic Computation, 38, 1535–1550.

    Article  MathSciNet  Google Scholar 

  • Leykin, A., & Tsai, H. Dmodules - Algorithms for \(D\)-modules. Macaulay2 package, http://www.math.uiuc.edu/Macaulay2/

  • Logar, A., & Sturmfels, B. (1992). Algorithms for the Quillen-Suslin theorem. Journal of Algebra, 145(1), 231–239.

    Article  MATH  MathSciNet  Google Scholar 

  • Lombardi, H., & Yengui, I. (2005). Suslin’s algorithms for reduction of unimodular rows. Journal of Symbolic Computation, 30(6), 707–717.

    Article  MathSciNet  Google Scholar 

  • Lu, P., Liu, M., & Oberst, U. (2004). Linear recurring arrays, linear systems and multidimensional cyclic codes over quasi-Frobenius rings. Acta Applicandae Mathematica, 80, 175–198.

    Article  MATH  MathSciNet  Google Scholar 

  • Malgrange, B. Systèmes différentiels à coefficients constants. Séminaire Bourbaki, 246, 1962–64, 11 pages.

    Google Scholar 

  • McConnell, J. C., & Robson, J. C. (2000). Noncommutative Noetherian Rings. American Mathematical Society Providence, RI. With the cooperation of L. W. Small. Revision of the 1987 edition.

  • Mounier, H. (1995). Propriétés structurelles des systèmes linéaires à retards: Aspects théoriques et pratiques. France: Ph.D. thesis, University of Orsay.

  • Mounier, H., Rudolph, J., Petitot, M., & Fliess, M. (1995). A flexible rod as a linear delay system. In Proceedings of the 3rd European Control Conference Italy: Rome.

  • Oberst, U. (1990). Multidimensional constant linear systems. Acta Applicandae Mathematica, 20, 1–175.

    Article  MATH  MathSciNet  Google Scholar 

  • Oldenburg, J., & Marquardt, W. (2002). Flatness and higher order differential model representations in dynamic optimization. Computers & Chemical Engineering, 26(3), 385–400.

    Article  Google Scholar 

  • Palamodov, V. P. (1970). Linear differential operators with constant coefficients. Translated from the Russian by A. A. Brown. Volume 168 of Grundlehren der mathematischen Wissenschaften. New York: Springer.

  • Park, H., & Woodburn, C. (1995). An algorithmic proof of Suslin’s stability theorem for polynomial rings. Journal of Algebra, 178(1), 277–298.

    Article  MATH  MathSciNet  Google Scholar 

  • Pillai, H. K., & Shankar, S. (1998). A behavioral approach to control of distributed systems. SIAM Journal on Control & Optimization, 37(2), 388–408.

    Article  MathSciNet  Google Scholar 

  • Polderman, J. W., & Willems, J. C. (1998). Introduction to mathematical systems theory. Volume 26 of Texts in applied mathematics. New York: Springer. A Behavioral Approach.

  • Pommaret, J.-F. (1978). Systems of partial differential equations and Lie pseudogroups. Volume 14 of mathematics and its applications. New York: Gordon & Breach Science Publishers.

  • Pommaret, J.-F. (1995). Dualité différentielle et applications. Comptes rendus de l’Académie des sciences, Paris. Série 1. Mathématique, 320, 1225–1230.

  • Pommaret, J.-F. (2001) Partial differential control theory. Volume 530 of Mathematics and its Applications. Dordrecht: Kluwer Academic Publishers Group. Vol. I. Mathematical tools; Vol. II. Control systems.

  • Pommaret, J.-F., & Quadrat, A. (1999a). Algebraic analysis of linear multidimensional control systems. IMA Journal of Mathematical Control and Information, 16(3), 275–297.

    Article  MATH  MathSciNet  Google Scholar 

  • Pommaret, J.-F., & Quadrat, A. (1999b). Localization and parametrization of linear multidimensional control systems. Systems & Control Letters, 37(4), 247–260.

    Article  MATH  MathSciNet  Google Scholar 

  • Pommaret, J.-F., & Quadrat, A. (2000). Equivalences of linear control systems. In Proceedings of MTNS 2000. France: Perpignan.

  • Pommaret, J.-F., & Quadrat A. (2003) A functorial approach to the behaviour of multidimensional control systems. International Journal of Applied Mathematics and Computer Science, 13(1), 7–13. Multidimensional systems \(n\text{ D }\) and iterative learning control (Czocha Castle, 2000).

    Google Scholar 

  • Pommaret, J.-F., & Quadrat, A. (2004). A differential operator approach to multidimensional optimal control. International Journal of Control, 77, 821–836.

    Article  MATH  MathSciNet  Google Scholar 

  • Quadrat, A. (1999). Analyse algébrique des systèmes de contrôle linéaires multidimensionnels. France: Ph.D. thesis, Ecole Nationale des Ponts et Chaussées.

  • Quadrat, A. (2010a). Systèmes et Structures: Une approche de la théorie mathématique des systèmes par l’analyse algébrique constructive. France: Habilitation thesis, Université de Nice Sophia Antipolis.

  • Quadrat, A. (2010b). An introduction to constructive algebraic analysis and its applications. In Les cours du CIRM, tome 1, numéro 2: Journées Nationales de Calcul Formel (pp. 281–471). Cf. also http://hal.archives-ouvertes.fr/inria-00506104/fr/

  • Quadrat, A. (2013). Grade filtration of linear functional systems. Acta Applicandae Mathematicae, 127, 27–86.

    Article  MATH  MathSciNet  Google Scholar 

  • Quadrat, A., & Regensburger, G. (2013). Polynomial solutions and annihilators of ordinary integro-differential operators. In Proceedings of the 5th Symposium on System Structure and Control (pp. 303–308). France: Grenoble.

  • Quadrat, A., & Robertz, D. (2005a). Parametrizing all solutions of uncontrollable multidimensional linear systems. In Proceedings of the 16th IFAC World Congress. Czech Republic: Prague.

  • Quadrat, A., & Robertz, D. (2005b). On the blowing-up of stably free behaviours. In Proceedings of the 44th IEEE Conference on Decision and Control and European Control Conference ECC 2005 (pp. 1541–1546). Spain: Seville.

  • Quadrat, A., & Robertz, D. (2006a). Constructive computation of flat outputs of a class of multidimensional linear systems with variable coefficients. In Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2006) (pp. 583–595). Kyoto: Japan.

  • Quadrat, A., & Robertz, D. (2006b). On the Monge problem and multidimensional optimal control. In Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2006) (pp. 596–605). Kyoto: Japan.

  • Quadrat, A., & Robertz, D. (2007). Computation of bases of free modules over the Weyl algebras. Journal of Symbolic Computation, 42(11–12), 1113–1141. Stafford project (http://wwwb.math.rwth-aachen.de/OreModules)

    Google Scholar 

  • Quadrat, A., & Robertz, D. (2008). Baer’s extension problem for multidimensional linear systems. In Proceedings of the 18th international symposium on mathematical theory of networks and systems (MTNS 2008). USA: Virginia Tech, Blacksburg, Virginia.

  • Quadrat, A., & Robertz, D. (2010). Controllability and differential flatness of linear analytic ordinary differential systems. In Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2010) Hungary: Budapest. Also in E. Zerz (eds.), Algebraic systems theory, behaviors, and codes (pp. 23–30). Shaker, Aachen.

  • Quadrat, A., & Robertz, D. (2013). Stafford’s reduction of linear partial differential systems. In Proceedings of the 5th Symposium on System Structure and Control (pp. 309–314). Grenoble: France.

  • Quadrat, A., & Robertz, D. (to appear). A constructive study of the module structure of rings of partial differential operators. Acta Applicandae Mathematicae.

  • Quillen, D. (1976). Projective modules over polynomial rings. Inventiones Mathematicae, 36, 167–171.

    Article  MATH  MathSciNet  Google Scholar 

  • Robertz, D. (2006). Formal computational methods for control theory. Ph.D. thesis, RWTH Aachen University, Germany. This thesis is available at http://darwin.bth.rwth-aachen.de/opus/volltexte/2006/1586

  • Robertz, D. (2007). Janet bases and applications. In M. Rosenkranz, D. Wang (Eds.) Gróbner bases in symbolic analysis. Volume 2 of Radon Series on Computational and Applied Mathematics (pp.139–168). Berlin: Walter de Gruyter.

  • Robertz, D. (2012). Formal Algorithmic Elimination for PDEs. Habilitationsschrift, accepted by the Faculty of Mathematics, Computer Science and Natural Sciences, RWTH Aachen University. Submitted for publication.

  • Rotman, J. J. (2009). An introduction to homological algebra (2nd ed.). New York: Universitext. Springer.

    Book  MATH  Google Scholar 

  • Serre, J.-P. (1958) Modules projectifs et espaces fibrés à fibre vectorielle. In Séminaire P. Dubreil, M.-L. (Eds.) Dubreil-Jacotin et C. Pisot, 1957/58, Fasc. 2, Exposé 23, Secrétariat mathématique (pp. 1–18).

  • Stafford, J. T. (1978). Module structure of Weyl algebras. Journal of the London Mathematical Society, 18, 429–442.

    Article  MATH  MathSciNet  Google Scholar 

  • Suslin, A. A. (1976). Projective modules over a polynomial ring are free. Dokl. Akad. Nauk. S.S.S.R., 229, 1063–1066.

    Google Scholar 

  • Wang, D. (2001). Elimination methods. Texts and monographs in symbolic computation. Vienna: Springer.

    Google Scholar 

  • Wood, J. (2000). Modules and behaviours in \(n\text{ D }\) systems theory. Multidimensional Systems and Signal Processing, 11(1–2), 11–48.

    Article  MATH  MathSciNet  Google Scholar 

  • Zervos, P. (1932). Le problème de Monge. Mémorial des Sciences Mathématiques, fasc. LIII. Gauthier-Villars.

  • Zerz, E. (2000). Topics in multidimensional linear systems theory. Volume 256 of Lecture Notes in Control and Information Sciences. London: Springer.

  • Zerz, E. (2001). Extension modules in behavioral linear systems theory. Multidimensional Systems and Signal Processing, 12(3–4), 309–327. Special issue: Applications of Gröbner bases to multidimensional systems and signal processing.

  • Zerz, E. (2006). An algebraic analysis approach to linear time-varying systems. IMA Journal of Mathematical Control & Information, 23, 113–126.

    Article  MATH  MathSciNet  Google Scholar 

  • Zerz, E., & Lomadze, V. (2001). A constructive solution to interconnection and decomposition problems with multidimensional behaviors. SIAM Journal on Control & Optimization, 40, 1072–1086.

    Article  MATH  MathSciNet  Google Scholar 

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Robertz, D. Recent progress in an algebraic analysis approach to linear systems. Multidim Syst Sign Process 26, 349–388 (2015). https://doi.org/10.1007/s11045-014-0280-9

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Keywords

  • Systems of linear functional equations
  • Systems theory
  • Control theory
  • Algebraic analysis
  • Homological algebra
  • Linear partial differential equations
  • Janet bases
  • Gröbner bases