Combinatorial Hopf Algebras for Interconnected Nonlinear Input-Output Systems with a View Towards Discretization

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 267)


A detailed expose of the Hopf algebra approach to interconnected input-output systems in nonlinear control theory is presented. The focus is on input-output systems that can be represented in terms of Chen–Fliess functional expansions or Fliess operators. This provides a starting point for a discrete-time version of this theory. In particular, the notion of a discrete-time Fliess operator is given and a class of parallel interconnections is described in terms of the quasi-shuffle algebra.


Nonlinear control systems Chen–Fliess series Combinatorial Hopf algebras 

AMS Subject Classification

93C10 93B25 16T05 16T30 



The third author was supported by grant SEV-2011-0087 from the Severo Ochoa Excellence Program at the Instituto de Ciencias Matemáticas in Madrid, Spain. This research was also supported by a grant from the BBVA Foundation.


  1. 1.
    Abe, E.: Hopf Algebras. Cambridge University Press, Cambridge (1980)zbMATHGoogle Scholar
  2. 2.
    Anshelevich, M., Effros, E.G., Popa, M.: Zimmerman type cancellation in the free Faà di Bruno algebra. J. Funct. Anal. 237, 76–104 (2006)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Berstel, J., Reutenauer, C.: Rational Series and Their Languages. Springer, Berlin (1988)CrossRefGoogle Scholar
  4. 4.
    Brockett, R.: Volterra series and geometric control theory. Automatica 12, 167–176 (1976), (addendum with E. Gilbert, 12, p. 635 (1976))MathSciNetCrossRefGoogle Scholar
  5. 5.
    Burde, D.: Left-symmetric algebras, or pre-Lie algebras in geometry and physics. Cent. Eur. J. Math. 4, 323–357 (2006)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Butler, S., Karasik, P.: A note on nested sums. J. Integer Seq. 13, article 10.4.4 (2010)Google Scholar
  7. 7.
    Cartier, P.: A primer of Hopf algebras. In: Cartier, P., Moussa, P., Julia, B., Vanhove, P. (eds.) Frontiers in Number Theory, Physics and Geometry II, pp. 537–615. Springer, Berlin Heidelberg (2007)CrossRefGoogle Scholar
  8. 8.
    Cartier, P.: Vinberg algebras, Lie groups and combinatorics. Clay Math. Proc. 11, 107–126 (2011)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Chen, K.T.: Iterated integrals and exponential homomorphisms. Proc. Lond. Math. Soc. 4, 502–512 (1954)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chen, K.T.: Algebraization of iterated integration along paths. Bull. AMS 73, 975–978 (1967)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chen, K.T.: Integration of paths, geometric invariants and a generalized Baker–Hausdorff formula. Ann. Math., 2nd Ser. 65, 163–178 (1975)Google Scholar
  12. 12.
    Devlin, J.: Word problems related to periodic solutions of a non-autonomous system. Math. Proc. Camb. Philos. Soc. 108, 127–151 (1990)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Duffaut Espinosa, L.A.: Interconnections of nonlinear systems driven by \(L_2\)-Itô stochastic processes. Dissertation, Old Dominion University, Norfolk, Virginia (2009)Google Scholar
  14. 14.
    Duffaut Espinosa, L.A., Ebrahimi-Fard, K., Gray, W.S.: A combinatorial Hopf algebra for nonlinear output feedback control systems. J. Algebra 453, 609–643 (2016). arXiv:1406.5396 MathSciNetCrossRefGoogle Scholar
  15. 15.
    Duffaut Espinosa, L.A., Gray, W.S.: Integration of output tracking and trajectory generation via analytic left inversion. In: Proceedings of the 21st International Conference on System Theory, Control and Computing, Sinaia, Romania, pp. 802–807 (2017)Google Scholar
  16. 16.
    Ebrahimi-Fard, K., Gray, W.S.: Center problem, Abel equation and the Faà di Bruno Hopf algebra for output feedback. Int. Math. Res. Not. 2017, 5415–5450 (2017). arXiv:1507.06939
  17. 17.
    Ebrahimi-Fard, K., Guo, L.: Mixable shuffles, quasi-shuffles and Hopf algebras. J. Algebraic Comb. 24, 83–101 (2006)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Ebrahimi-Fard, K., Patras, F.: La structure combinatoire du calcul intégral. Gaz. Math. 138, 5–22 (2013)Google Scholar
  19. 19.
    Ferfera, A.: Combinatoire du monoïde libre appliquée à la composition et aux variations de certaines fonctionnelles issues de la théorie des systèmes. Dissertation, University of Bordeaux I, Talence (1979)Google Scholar
  20. 20.
    Ferfera, A.: Combinatoire du monoïde libre et composition de certains systèmes non linéaires. Astérisque 75–76, 87–93 (1980)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Figueroa, H., Gracia-Bondía, J.M.: Combinatorial Hopf algebras in quantum field theory I. Rev. Math. Phys. 17, 881–976 (2005)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Figueroa, H., Gracia-Bondía, J.M., Várilly, J.C.: Elements of Noncommutative Geometry, Birkhäuser, Boston (2001)Google Scholar
  23. 23.
    Fliess, M.: Generating series for discrete-time nonlinear systems. IEEE Trans. Autom. Control AC-25, 984–985 (1980)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Fliess, M.: Fonctionnelles causales non linéaires et indéterminées non commutatives. Bull. Soc. Math. Fr. 109, 3–40 (1981)CrossRefGoogle Scholar
  25. 25.
    Fliess, M.: Réalisation locale des systèmes non linéaires, algèbres de Lie filtrées transitives et séries génératrices non commutatives. Invent. Math. 71, 521–537 (1983)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Foissy, L.: The Hopf algebra of Fliess operators and its dual pre-Lie algebra. Commun. Algebra 43, 4528–4552 (2015). arXiv:0805.4385v2 MathSciNetCrossRefGoogle Scholar
  27. 27.
    Frabetti, A., Manchon, D.: Five interpretations of Faà di Bruno’s formula. In: Ebrahimi-Fard, K., Fauvet, F. (eds.) Faà di Bruno Hopf Algebras, Dyson-Schwinger Equations, and Lie-Butcher Series. IRMA Lectures in Mathematics and Theoretical Physics, vol. 21, pp. 91–147. European Mathematical Society, Zürich, Switzerland (2015)Google Scholar
  28. 28.
    Gray, W.S., Duffaut Espinosa, L.A.: A Faà di Bruno Hopf algebra for a group of Fliess operators with applications to feedback. Syst. Control Lett. 60, 441–449 (2011)CrossRefGoogle Scholar
  29. 29.
    Gray, W.S., Duffaut Espinosa, L.A.: Feedback transformation groups for nonlinear input-output systems. In: Proceedings of the 52nd IEEE Conference on Decision and Control, Florence, Italy, pp. 2570–2575 (2013)Google Scholar
  30. 30.
    Gray, W.S., Duffaut Espinosa, L.A.: A Faà di Bruno Hopf algebra for analytic nonlinear feedback control systems. In: Ebrahimi-Fard, K., Fauvet, F. (eds.) Faà di Bruno Hopf Algebras, Dyson-Schwinger Equations, and Lie-Butcher Series. IRMA Lectures in Mathematics and Theoretical Physics, vol. 21, pp. 149–217. European Mathematical Society, Zürich, Switzerland (2015)Google Scholar
  31. 31.
    Gray, W.S., Duffaut Espinosa, L.A., Ebrahimi-Fard, K.: Recursive algorithm for the antipode in the SISO feedback product. In: Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems, Groningen, The Netherlands, pp. 1088–1093 (2014)Google Scholar
  32. 32.
    Gray, W.S., Duffaut Espinosa, L.A., Ebrahimi-Fard, K.: Faà di Bruno Hopf algebra of the output feedback group for multivariable Fliess operators. Syst. Control Lett. 74, 64–73 (2014)CrossRefGoogle Scholar
  33. 33.
    Gray, W.S., Duffaut Espinosa, L.A., Ebrahimi-Fard, K.: Analytic left inversion of SISO Lotka-Volterra models. In: Proceedings of the 49th Conference on Information Sciences and Systems, Baltimore, Maryland (2015)Google Scholar
  34. 34.
    Gray, W.S., Duffaut Espinosa, L.A., Ebrahimi-Fard, K.: Analytic left inversion of multivariable Lotka-Volterra models. In: Proceedings of the 54th IEEE Conference on Decision and Control, Osaka, Japan, pp. 6472–6477 (2015)Google Scholar
  35. 35.
    Gray, W.S., Duffaut Espinosa, L.A., Ebrahimi-Fard, K.: Discrete-time approximations of Fliess operators. In: Proceedings of the American Control Conference. Boston, Massachusetts, pp. 2433–2439 (2016)Google Scholar
  36. 36.
    Gray, W.S., Duffaut Espinosa, L.A., Ebrahimi-Fard, K.: Discrete-time approximations of Fliess operators. Numerische Mathematik 137, 35–62 (2017). arXiv:1510.07901 MathSciNetCrossRefGoogle Scholar
  37. 37.
    Gray, W.S., Duffaut Espinosa, L.A., Thitsa, M.: Left inversion of analytic nonlinear SISO systems via formal power series methods. Automatica 50, 2381–2388 (2014)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Gray, W.S., Li, Y.: Generating series for interconnected analytic nonlinear systems. SIAM J. Control Optim. 44, 646–672 (2005)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Gray, W.S., Thitsa, M.: A unified approach to generating series for cascaded nonlinear input-output systems. Int. J. Control 85, 1737–1754 (2012)CrossRefGoogle Scholar
  40. 40.
    Gray, W.S., Wang, Y.: Fliess operators on \(L_p\) spaces: convergence and continuity. Syst. Control Lett. 46, 67–74 (2002)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Hoffman, M.E.: Quasi-shuffle products. J. Algebraic Comb. 11, 49–68 (2000)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Houseaux, V., Jacob, G., Oussous, N.E., Petitot, M.: A complete Maple package for noncommutative rational power series. In: Li, Z., Sit, W.Y. (eds.) Computer Mathematics: Proceeding of the Sixth Asian Symposium. Lecture Notes Series on Computing, vol. 10, pp. 174–188. World Scientific, Singapore (2003)CrossRefGoogle Scholar
  43. 43.
    Isidori, A.: Nonlinear Control Systems, 3rd edn. Springer, London (1995)CrossRefGoogle Scholar
  44. 44.
    Jakubczyk, B.: Existence and uniqueness of realizations of nonlinear systems. SIAM J. Control Optim. 18, 445–471 (1980)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Jakubczyk, B.: Local realization of nonlinear causal operators. SIAM J. Control Optim. 24, 230–242 (1986)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Jakubczyk, B.: Realization theory for nonlinear systems: three approaches. In: Fliess, M., Hazewinkel M. (eds.) Algebraic and Geometric Methods in Nonlinear Control Theory, pp. 3–31. D. Reidel Publishing Company, Dordrecht (1986)CrossRefGoogle Scholar
  47. 47.
    Khalil, H.K.: Nonlinear Systems, 3rd edn. Prentice-Hall, Upper Saddle River, New Jersey (2002)Google Scholar
  48. 48.
    Lesiak, C., Krener, A.J.: The existence and uniqueness of Volterra series for nonlinear systems. IEEE Trans. Autom. Control AC-23, 1090–1095 (1978)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Li, Y.: Generating series of interconnected nonlinear systems and the formal Laplace-Borel transform. Dissertation, Old Dominion University, Norfolk, Virginia (2004)Google Scholar
  50. 50.
    Manchon, D.: Hopf algebras, from basics to applications to renormalization. Comptes-rendus des Rencontres mathématiques de Glanon 2001 (2001). arXiv:0408405
  51. 51.
    Manchon, D.: A short survey on pre-Lie algebras. In: Carey, A. (ed.) Noncommutative Geometry and Physics: Renormalisation, Motives, Index Theory, pp. 89–102. European Mathematical Society, Zürich, Switzerland (2011)Google Scholar
  52. 52.
    Milnor, J.W., Moore, J.C.: On the structure of Hopf algebras. Ann. Math. 81, 211–264 (1965)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Nijmeijer, H., van der Schaft, A.J.: Nonlinear Dynamical Control Systems. Springer, New York (1990)CrossRefGoogle Scholar
  54. 54.
    Pittou, M., Rahonis, G.: Weighted recognizability over infinite alphabets. Acta Cybern. 23, 283–317 (2017)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Radford, D.E.: Hopf Algebras. World Scientific Publishing, Hackensack, New Jersey (2012)Google Scholar
  56. 56.
    Ree, R.: Lie elements and an algebra associated with shuffles. Ann. Math., 2nd Ser. 68, 210–220 (1958)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Reutenauer, C.: Free Lie Algebras. Oxford University Press, New York (1993)zbMATHGoogle Scholar
  58. 58.
    Rugh, W.J.: Nonlinear System Theory, The Volterra/Wiener Approach. The Johns Hopkins University Press, Baltimore, Maryland (1981)zbMATHGoogle Scholar
  59. 59.
    Schützenberger, M.P.: On the definition of a family of automata. Inf. Control 4, 245–270 (1961)MathSciNetCrossRefGoogle Scholar
  60. 60.
    Slotine, J.-J.E., Li, W.: Applied Nonlinear Control. Prentice-Hall, Englewood Cliffs, New Jersey (1991)Google Scholar
  61. 61.
    Sontag, E.D.: Polynomial Response Maps. Springer, Berlin (1979)CrossRefGoogle Scholar
  62. 62.
    Sussmann, H.J.: Existence and uniqueness of minimal realizations of nonlinear systems. Math. Syst. Theory 10, 263–284 (1977)MathSciNetCrossRefGoogle Scholar
  63. 63.
    Sussmann, H.J.: A proof of the realization theorem for convergent generating series of finite Lie rank, internal report SYCON-90-02, Rutgers Center for Systems and Control (1990)Google Scholar
  64. 64.
    Sweedler, M.E.: Hopf Algebras. W. A. Benjamin Inc., New York (1969)zbMATHGoogle Scholar
  65. 65.
    Thitsa, M., Gray, W.S.: On the radius of convergence of interconnected analytic nonlinear input-output systems. SIAM J. Control Optim. 50, 2786–2813 (2012)MathSciNetCrossRefGoogle Scholar
  66. 66.
    Wang, Y.: Differential equations and nonlinear control systems. Dissertation, Rutgers University, New Brunswick, New Jersey (1990)Google Scholar

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Authors and Affiliations

  1. 1.Department of Electrical and Biomedical EngineeringUniversity of VermontBurlingtonUSA
  2. 2.Department of Mathematical SciencesNorwegian University of Science and Technology – NTNUTrondheimNorway
  3. 3.Department of Electrical and Computer EngineeringOld Dominion UniversityNorfolkUSA

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