Recent progress in an algebraic analysis approach to linear systems

  • D. Robertz


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.


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


  1. 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
  2. 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.Google Scholar
  3. 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.Google Scholar
  4. Barakat, M., & Robertz, D. (2008). homalg: A meta-package for homological algebra. Journal of Algebra and its Applications, 7(3), 299–317.CrossRefMATHMathSciNetGoogle Scholar
  5. Björk, J.-E. (1979). Rings of differential operators. Volume 21 of North-Holland Mathematical Library. Amsterdam: North-Holland Publishing Co.Google Scholar
  6. 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.
  7. Boudellioua, M. S., & Quadrat, A. (2010). Serre’s reduction of linear functional systems. Mathematics in Computer Science, 4(2–3), 289–312.CrossRefMATHMathSciNetGoogle Scholar
  8. 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.CrossRefMATHMathSciNetGoogle Scholar
  9. Bourbaki, N. (1980). Algèbre, Chapter 10. Algèbre homologique. Paris: Masson.Google Scholar
  10. 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.CrossRefMATHMathSciNetGoogle Scholar
  11. 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
  12. Caro, N., & Levcovitz, D. (2010). On a Theorem of Stafford. Cadernos de Mathemática, 11, 63–70.Google Scholar
  13. 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.Google Scholar
  14. Chyzak, F. (1998). Fonctions holonomes en calcul formel. Palaiseau, France: Ph.D. thesis, Ecole Polytechnique.Google Scholar
  15. 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.CrossRefMATHMathSciNetGoogle Scholar
  16. 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
  17. Chyzak, F., & Salvy, B. (1998). Non-commutative elimination in Ore algebras proves multivariate identities. Journal of Symbolic Computation, 26, 187–227.CrossRefMATHMathSciNetGoogle Scholar
  18. Cluzeau, T., & Quadrat, A. (2008). Factoring and decomposing a class of linear functional systems. Linear Algebra and Its Applications, 428(1), 324–381.CrossRefMATHMathSciNetGoogle Scholar
  19. 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 or
  20. Cluzeau, T., & Quadrat, A. (2012). Serre’s reduction of linear partial differential systems with holonomic adjoints. Journal of Symbolic Computation, 47(10), 1192–1213.CrossRefMATHMathSciNetGoogle Scholar
  21. Cohn, P. M. (2006). Free ideal rings and localization in general rings. Volume 3 of New Mathematical Monographs. Cambridge: Cambridge University Press.Google Scholar
  22. Coutinho, S. C., & Holland, M. P. (1988). Module structure of rings of differential operators. Proceedings of the London Mathematical Society, 57, 417–432.CrossRefMATHMathSciNetGoogle Scholar
  23. 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).Google Scholar
  24. Ehrenpreis, L. (1970). Fourier analysis in several complex variables. Volume XVII of Pure and Applied Mathematics. New York-London-Sydney: Wiley-Interscience Publishers, Wiley.Google Scholar
  25. Eisenbud, D. (1995). Commutative algebra. Volume 150 of Graduate Texts in Mathematics. New York: Springer. With a view toward algebraic geometry.Google Scholar
  26. Fabiańska, A. (2009). Algorithmic analysis of presentations of groups and modules. Germany: Ph.D. thesis, RWTH Aachen University. This thesis is available at QuillenSuslin project (
  27. 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.Google Scholar
  28. 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.Google Scholar
  29. Fliess, M. (1991). Controllability revisited. In A. C. Antoulas (Ed.), Mathematical system theory (pp. 463–474). Berlin: Springer. The influence of R. E. Kalman.CrossRefGoogle Scholar
  30. 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.CrossRefMATHMathSciNetGoogle Scholar
  31. 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.CrossRefMATHMathSciNetGoogle Scholar
  32. Freund, E. (1971). Zeitvariable Mehrgrößensysteme. Volume 57 of Lecture Notes in Operations Research and Mathematical Systems. Heidelberg: Springer.Google Scholar
  33. Fröhler, S., & Oberst, U. (1998). Continuous time-varying linear systems. Systems & Control Letters, 35(2), 97–110.CrossRefMATHMathSciNetGoogle Scholar
  34. Gago-Vargas, J. (2003). Bases for projective modules in \(A_n(k)\). Journal of Symbolic Computation, 36(6), 845–853.CrossRefMATHMathSciNetGoogle Scholar
  35. Gallego, C., & Lezama, O. (2011). Gröbner bases for ideals of \(\sigma \text{- }PBW\) extensions. Communications in Algebra, 39(1), 50–75.CrossRefMATHMathSciNetGoogle Scholar
  36. 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
  37. Gluesing-Luerssen, H. (2002). Linear delay-differential systems with commensurate delays: An algebraic approach. Volume 1770 of Lecture Notes in Mathematics. Berlin: Springer.Google Scholar
  38. 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.MATHGoogle Scholar
  39. Guo, L., Regensburger, G., & Rosenkranz, M. (2014). On integro-differential algebras. Journal of Pure and Applied Algebra, 218(3), 456–473.CrossRefMATHMathSciNetGoogle Scholar
  40. Hillebrand, A., & Schmale, W. (2001). Towards an effective version of a theorem of Stafford. Journal of Symbolic Computation, 32, 699–716.CrossRefMATHMathSciNetGoogle Scholar
  41. Janet, M. (1929). Leçons sur les systèmes d’équations aux dérivées partielles. Cahiers Scientifiques, fasc. IV. Paris: Gauthiers-Villars.Google Scholar
  42. Janet, M. (1971). P. Zervos et le problème de Monge. Bull. Sci. Math. (2), 95, 15–26.Google Scholar
  43. Kandri-Rody, A., & Weispfenning, V. (1990). Noncommutative Gröbner bases in algebras of solvable type. Journal of Symbolic Computation, 9(1), 1–26.CrossRefMATHMathSciNetGoogle Scholar
  44. Kashiwara, M. (1995). Algebraic study of systems of partial differential equations, Volume 63. France (N.S.): Mém. Soc. Math.Google Scholar
  45. Kredel, H. (1993). Solvable polynomial rings. Aachen: Shaker.MATHGoogle Scholar
  46. Lam, T. Y. (1999). Lectures on modules and rings. Volume 189 of Graduate Texts in Mathematics. New York: Springer.Google Scholar
  47. Lam, T. Y. (2006). Serre’s problem on projective modules. Springer Monographs in Mathematics. Berlin: Springer.CrossRefGoogle Scholar
  48. 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.Google Scholar
  49. 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
  50. 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
  51. Lévine, J. (2011). On necessary and sufficient conditions for differential flatness. Applicable Algebra in Engineering, Communication and Computing, 22(1), 47–90.CrossRefMATHMathSciNetGoogle Scholar
  52. Leykin, A. (2004). Algorithmic proofs of two theorems of Stafford. Journal of Symbolic Computation, 38, 1535–1550.CrossRefMathSciNetGoogle Scholar
  53. Leykin, A., & Tsai, H. Dmodules - Algorithms for \(D\)-modules. Macaulay2 package,
  54. Logar, A., & Sturmfels, B. (1992). Algorithms for the Quillen-Suslin theorem. Journal of Algebra, 145(1), 231–239.CrossRefMATHMathSciNetGoogle Scholar
  55. Lombardi, H., & Yengui, I. (2005). Suslin’s algorithms for reduction of unimodular rows. Journal of Symbolic Computation, 30(6), 707–717.CrossRefMathSciNetGoogle Scholar
  56. 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.CrossRefMATHMathSciNetGoogle Scholar
  57. Malgrange, B. Systèmes différentiels à coefficients constants. Séminaire Bourbaki, 246, 1962–64, 11 pages.Google Scholar
  58. 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.Google Scholar
  59. Mounier, H. (1995). Propriétés structurelles des systèmes linéaires à retards: Aspects théoriques et pratiques. France: Ph.D. thesis, University of Orsay.Google Scholar
  60. 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.Google Scholar
  61. Oberst, U. (1990). Multidimensional constant linear systems. Acta Applicandae Mathematica, 20, 1–175.CrossRefMATHMathSciNetGoogle Scholar
  62. Oldenburg, J., & Marquardt, W. (2002). Flatness and higher order differential model representations in dynamic optimization. Computers & Chemical Engineering, 26(3), 385–400.CrossRefGoogle Scholar
  63. 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.Google Scholar
  64. Park, H., & Woodburn, C. (1995). An algorithmic proof of Suslin’s stability theorem for polynomial rings. Journal of Algebra, 178(1), 277–298.CrossRefMATHMathSciNetGoogle Scholar
  65. Pillai, H. K., & Shankar, S. (1998). A behavioral approach to control of distributed systems. SIAM Journal on Control & Optimization, 37(2), 388–408.CrossRefMathSciNetGoogle Scholar
  66. 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.Google Scholar
  67. 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.Google Scholar
  68. Pommaret, J.-F. (1995). Dualité différentielle et applications. Comptes rendus de l’Académie des sciences, Paris. Série 1. Mathématique, 320, 1225–1230.Google Scholar
  69. 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.Google Scholar
  70. Pommaret, J.-F., & Quadrat, A. (1999a). Algebraic analysis of linear multidimensional control systems. IMA Journal of Mathematical Control and Information, 16(3), 275–297.CrossRefMATHMathSciNetGoogle Scholar
  71. Pommaret, J.-F., & Quadrat, A. (1999b). Localization and parametrization of linear multidimensional control systems. Systems & Control Letters, 37(4), 247–260.CrossRefMATHMathSciNetGoogle Scholar
  72. Pommaret, J.-F., & Quadrat, A. (2000). Equivalences of linear control systems. In Proceedings of MTNS 2000. France: Perpignan.Google Scholar
  73. 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
  74. Pommaret, J.-F., & Quadrat, A. (2004). A differential operator approach to multidimensional optimal control. International Journal of Control, 77, 821–836.CrossRefMATHMathSciNetGoogle Scholar
  75. Quadrat, A. (1999). Analyse algébrique des systèmes de contrôle linéaires multidimensionnels. France: Ph.D. thesis, Ecole Nationale des Ponts et Chaussées.Google Scholar
  76. 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.Google Scholar
  77. 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
  78. Quadrat, A. (2013). Grade filtration of linear functional systems. Acta Applicandae Mathematicae, 127, 27–86.CrossRefMATHMathSciNetGoogle Scholar
  79. 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.Google Scholar
  80. Quadrat, A., & Robertz, D. (2005a). Parametrizing all solutions of uncontrollable multidimensional linear systems. In Proceedings of the 16th IFAC World Congress. Czech Republic: Prague.Google Scholar
  81. 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.Google Scholar
  82. 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.Google Scholar
  83. 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.Google Scholar
  84. 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 ( Scholar
  85. 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.Google Scholar
  86. 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.Google Scholar
  87. 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.Google Scholar
  88. Quadrat, A., & Robertz, D. (to appear). A constructive study of the module structure of rings of partial differential operators. Acta Applicandae Mathematicae.Google Scholar
  89. Quillen, D. (1976). Projective modules over polynomial rings. Inventiones Mathematicae, 36, 167–171.CrossRefMATHMathSciNetGoogle Scholar
  90. Robertz, D. (2006). Formal computational methods for control theory. Ph.D. thesis, RWTH Aachen University, Germany. This thesis is available at
  91. 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.Google Scholar
  92. 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.Google Scholar
  93. Rotman, J. J. (2009). An introduction to homological algebra (2nd ed.). New York: Universitext. Springer.CrossRefMATHGoogle Scholar
  94. 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).Google Scholar
  95. Stafford, J. T. (1978). Module structure of Weyl algebras. Journal of the London Mathematical Society, 18, 429–442.CrossRefMATHMathSciNetGoogle Scholar
  96. Suslin, A. A. (1976). Projective modules over a polynomial ring are free. Dokl. Akad. Nauk. S.S.S.R., 229, 1063–1066.Google Scholar
  97. Wang, D. (2001). Elimination methods. Texts and monographs in symbolic computation. Vienna: Springer.Google Scholar
  98. Wood, J. (2000). Modules and behaviours in \(n\text{ D }\) systems theory. Multidimensional Systems and Signal Processing, 11(1–2), 11–48.CrossRefMATHMathSciNetGoogle Scholar
  99. Zervos, P. (1932). Le problème de Monge. Mémorial des Sciences Mathématiques, fasc. LIII. Gauthier-Villars.Google Scholar
  100. Zerz, E. (2000). Topics in multidimensional linear systems theory. Volume 256 of Lecture Notes in Control and Information Sciences. London: Springer.Google Scholar
  101. 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.Google Scholar
  102. Zerz, E. (2006). An algebraic analysis approach to linear time-varying systems. IMA Journal of Mathematical Control & Information, 23, 113–126.CrossRefMATHMathSciNetGoogle Scholar
  103. Zerz, E., & Lomadze, V. (2001). A constructive solution to interconnection and decomposition problems with multidimensional behaviors. SIAM Journal on Control & Optimization, 40, 1072–1086.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of Computing and MathematicsPlymouth UniversityPlymouth, United Kingdom

Personalised recommendations