### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 082, 15 pages      arXiv:2202.10170      https://doi.org/10.3842/SIGMA.2022.082
Contribution to the Special Issue on Non-Commutative Algebra, Probability and Analysis in Action

### Entropy of Generating Series for Nonlinear Input-Output Systems and Their Interconnections

W. Steven Gray
Department of Electrical and Computer Engineering, Old Dominion University, Norfolk, Virginia 23529, USA

Received February 22, 2022, in final form October 07, 2022; Published online October 25, 2022

Abstract
This paper has two main objectives. The first is to introduce a notion of entropy that is well suited for the analysis of nonlinear input-output systems that have a Chen-Fliess series representation. The latter is defined in terms of its generating series over a noncommutative alphabet. The idea is to assign an entropy to a generating series as an element of a graded vector space. The second objective is to describe the entropy of generating series originating from interconnected systems of Chen-Fliess series that arise in the context of control theory. It is shown that one set of interconnections can never increase entropy as defined here, while a second set has the potential to do so. The paper concludes with a brief introduction to an entropy ultrametric space and some open questions.

Key words: Chen-Fliess series; formal power series; entropy; nonlinear control theory.

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References

1. Anick D.J., Non-commutative graded algebras and their Hilbert series, J. Algebra 78 (1982), 120-140.
2. Atiyah M.F., MacDonald I.G., Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass. - London - Don Mills, Ont., 1969.
3. Berstel J., Reutenauer C., Rational series and their languages, EATCS Monogr. Theoret. Comput. Sci., Vol. 12, Springer-Verlag, Berlin, 1988.
4. Caracciolo S., Radicati L.A., Entropy ultrametric for dynamical and disordered systems, J. Physique 50 (1989), 2919-2930.
5. Devlin J., Word problems related to periodic solutions of a nonautonomous system, Math. Proc. Cambridge Philos. Soc. 108 (1990), 127-151.
6. Devlin J., Word problems related to derivatives of the displacement map, Math. Proc. Cambridge Philos. Soc. 110 (1991), 569-579.
7. Downarowicz T., Entropy in dynamical systems, New Math. Monogr., Vol. 18, Cambridge University Press, Cambridge, 2011.
8. Duffaut Espinosa L.A., Interconnections of nonlinear systems driven by $L_2$-Itô stochastic process, Ph.D. Thesis, Old Dominion University, 2009, https://doi.org/10.25777/cngz-5y11.
9. Duffaut Espinosa L.A., Ebrahimi-Fard K., Gray W.S., A combinatorial Hopf algebra for nonlinear output feedback control systems, J. Algebra 453 (2016), 609-643, arXiv:1406.5396.
10. Duffaut Espinosa L.A., Gray W.S., González O.R., On Fliess operators driven by $L_2$-Itô random processes, in Proc. 48th IEEE Conf. on Decision and Control, IEEE, Shanghai, China, 2009, 7478-7484.
11. 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 (2017), 5415-5450, arXiv:1507.06939.
12. Ferfera A., Combinatoire du monoïde libre appliquée à la composition et aux variations de certaines fonctionnelles issues de la théorie des systèmes, Ph.D. Thesis, University of Bordeaux I, 1979.
13. Ferfera A., Combinatoire du monoïde libre et composition de certains systèmes non linéaires, Astérisque 75-76 (1980), 87-93, available at http://www.numdam.org/item/AST_1980__75-76__87_0.
14. Fliess M., Sur divers produits de séries formelles, Bull. Soc. Math. France 102 (1974), 181-191.
15. Fliess M., Transductions de séries formelles, Discrete Math. 10 (1974), 57-74.
16. Fliess M., Fonctionnelles causales non linéaires et indéterminées non commutatives, Bull. Soc. Math. France 109 (1981), 3-40.
17. Fliess M., Réalisation locale des systèmes non linéaires, algèbres de Lie filtrées transitives et séries génératrices non commutatives, Invent. Math. 71 (1983), 521-537.
18. Foissy L., The Hopf algebra of Fliess operators and its dual pre-Lie algebra, Comm. Algebra 43 (2015), 4528-4552, arXiv:1304.1726.
19. Gray W.S., Formal power series methods in nonlinear control theory, unpublished, 1.2 ed., 2022, available at http://www.ece.odu.edu/ sgray/fps-book.
20. Gray W.S., System identification entropy for Chen-Fliess series and their interconnections, in Proc. 2022 Allerton Conf. on Communication, Control, & Computing, Allerton Park, Illinois, to appear.
21. Gray W.S., Duffaut Espinosa L.A., Ebrahimi-Fard K., Faà di Bruno Hopf algebra of the output feedback group for multivariable Fliess operators, Systems Control Lett. 74 (2014), 64-73, arXiv:1406.5378.
22. Gray W.S., Ebrahimi-Fard K., SISO output affine feedback transformation group and its Faà di Bruno Hopf algebra, SIAM J. Control Optim. 55 (2017), 885-912, arXiv:1411.0222.
23. Gray W.S., Ebrahimi-Fard K., Generating series for networks of Chen-Fliess series, Systems Control Lett. 147 (2021), 104827, 8 pages, arXiv:2007.00743.
24. Gray W.S., Herencia-Zapana H., Duffaut Espinosa L.A., González O.R., Bilinear system interconnections and generating series of weighted Petri nets, Systems Control Lett. 58 (2009), 841-848.
25. Gray W.S., Li Y., Generating series for interconnected analytic nonlinear systems, SIAM J. Control Optim. 44 (2005), 646-672.
26. Gray W.S., Thitsa M., A unified approach to generating series for mixed cascades of analytic nonlinear input-output systems, Internat. J. Control 85 (2012), 1737-1754.
27. Gray W.S., Venkatesh G.S., Relative degree of interconnected SISO nonlinear control systems, Systems Control Lett. 124 (2019), 99-105.
28. Gray W.S., Wang Y., Fliess operators on $L_p$ spaces: convergence and continuity, Systems Control Lett. 46 (2002), 67-74.
29. Gray W.S., Wang Y., Formal Fliess operators with applications to feedback interconnections, in Proc. 18th Inter. Symp. on the Mathematical Theory of Networks and Systems, Blacksburg, Virginia, 2008, 12 pages.
30. Isidori A., Nonlinear control systems, 3rd ed., Comm. Control Engrg. Ser., Springer-Verlag, Berlin, 1995.
31. Jacob G., Sur un théorème de Shamir, Inf. Control 27 (1975), 218-261.
32. Kawan C., Delvenne J.-C., Network entropy and data rates required for networked control, IEEE Trans. Control Netw. Syst. 3 (2016), 57-66, arXiv:1409.6037.
33. Kuich W., On the entropy of context-free languages, Inf. Control 16 (1970), 173-200.
34. Kuich W., Maurer H., The structure generating function and entropy of tuple languages, Inf. Control 19 (1971), 195-203.
35. Kuich W., Salomaa A., Semirings, automata, languages, EATCS Monogr. Theoret. Comput. Sci., Vol. 5, Springer-Verlag, Berlin, 1986.
36. Liberzon D., On topological entropy of interconnected nonlinear systems, IEEE Control Syst. Lett. 5 (2021), 2210-2214.
37. Lind D., Marcus B., An introduction to symbolic dynamics and coding, 2nd ed., Cambridge Math. Lib., Cambridge University Press, Cambridge, 2021.
38. Lothaire M., Combinatorics on words, 2nd ed., Cambridge Math. Lib., Cambridge University Press, Cambridge, 1997.
39. Matveev A.S., Proskurnikov A.V., Pogromsky A., Fridman E., Comprehending complexity: data-rate constraints in large-scale networks, IEEE Trans. Automat. Control 64 (2019), 4252-4259.
40. McLachlan R.I., Ryland B., The algebraic entropy of classical mechanics, J. Math. Phys. 44 (2003), 3071-3087, arXiv:math-ph/0210030.
41. Newman M.F., Schneider C., Shalev A., The entropy of graded algebras, J. Algebra 223 (2000), 85-100.
42. Priess-Crampe S., Ribenboim P., Ultrametric dynamics, Illinois J. Math. 55 (2011), 287-303.
43. Priess-Crampe S., Ribenboim P., The approximation to a fixed point, J. Fixed Point Theory Appl. 14 (2013), 41-53, arXiv:1307.6431.
44. Salomaa A., Soittola M., Automata-theoretic aspects of formal power series, Monogr. Comput. Sci., Springer-Verlag, New York - Heidelberg, 1978.
45. Savkin A.V., Analysis and synthesis of networked control systems: topological entropy, observability, robustness and optimal control, Automatica 42 (2006), 51-62.
46. Schneider F.M., Borchmann D., Topological entropy of formal languages, Semigroup Forum 94 (2017), 556-581, arXiv:1507.03393.
47. Shannon C.E., A mathematical theory of communication, Bell System Tech. J. 27 (1948), 379-423.
48. Sloane N.J.A., The on-line encyclopedia of integer sequences, available at https://oeis.org.
49. Smith C., Enumeration of the distinct shuffles of permutations, Ph.D. Thesis, Harvard University, 2009, available at https://www.proquest.com/docview/304890446.
50. Thitsa M., Gray W.S., On the radius of convergence of interconnected analytic nonlinear input-output systems, SIAM J. Control Optim. 50 (2012), 2786-2813.
51. Tomar M.S., Zamani M., Compositional quantification of invariance feedback entropy for networks of uncertain control systems, IEEE Control Syst. Lett. 4 (2020), 827-832.
52. Venkatesh G.S., Wiener-Fliess composition of formal power series: additive static feedback and shuffle rational series, Ph.D. Thesis, Old Dominion University, 2021, https://doi.org/10.25777/t2b1-dx91.
53. Venkatesh G.S., Gray W.S., Formal power series approach to nonlinear systems with static output feedback, Internat. J. Control, to appear, arXiv:2110.10034.
54. Wang Y., Algebraic differential equations and nonlinear control systems, Ph.D. Thesis, Rutgers University, 1990, available at https://www.proquest.com/docview/303872790.
55. Winter-Arboleda I.M., Gray W.S., Duffaut Espinosa L.A., Fractional Fliess operators: two approaches, in Proc. 49th Conf. on Information Sciences and Systems, IEEE, Baltimore, Maryland, 2015, 6 pages.
56. Young L.-S., Entropy in dynamical systems, in Entropy, Princeton Ser. Appl. Math., Princeton University Press, Princeton, New Jersey, 2003, 313-327.
57. Zames G., On the metric complexity of causal linear systems: $\varepsilon$-entropy and $\varepsilon$-dimension for continuous time, IEEE Trans. Automat. Control 24 (1979), 222-230.