Homogeneous approximation for minimal realizations of series of iterated integrals

Keywords: homogeneous approximation, series of iterated integrals, minimal realization, core Lie subalgebra

Abstract

In the paper, realizable series of iterated integrals with scalar coefficients are considered and an algebraic approach to the homogeneous approximation problem for nonlinear control systems with output is developed. In the first section we recall the concept of the homogeneous approximation of a nonlinear control system which is linear w.r.t.\ the control and the concept of the series of iterated integrals. In the second section the statement of the realizability problem is given, a criterion for realizability and a method for constructing a minimal realization of the series are recalled. Also we recall some ideas of the algebraic approach to the description of the homogeneous approximation: the free graded associative algebra, which is isomorphic to the algebra of iterated integrals, the free Lie algebra, the Poincar\'{e}-Birkhoff-Witt basis, the dual basis and its construction by use of the shuffle product, the definition of the core Lie subalgebra, which defines the homogeneous approximation of a control system. In the third section we show how to find the core Lie subalgebra of the systems that is a realization of the one-dimensional series of iterated integrals without finding the system itself. The result obtained is illustrated by the example, in which we demonstrate two methods for finding the core Lie subalgebra of the realizing system. In the last section it is shown that for any graded Lie subalgebra of finite codimension there exists a one-dimensional homogeneous series such that this Lie subalgebra is the core Lie subalgebra for its minimal realization. The proof is constructive: we give a method of finding such a series; we use the dual basis to the Poincar\'{e}-Birkhoff-Witt basis of the free associative algebra, which is built by the core Lie subalgebra, and the shuffle product in this algebra. As a consequence, we get a classification of all possible homogeneous approximations of systems that are realizations of one-dimensional series of iterated integrals.

References

A. A. Agrachev, R. V. Gamkrelidze, A. V. Sarychev. Local invariants of smooth control systems, Acta Appl. Math. -- 1989. - Vol. 14. - P.191-237. DOI: http://doi.org/10.1007/BF01307214

A. Bellaiche. The tangent space in sub-Riemannian geometry, in: Progress in Mathematics, Bellaiche, A. and Risler, J. J., eds., Birkh"{a}user Basel, 1996. -- Vol. 144. -- P.~1--78. DOI: http://doi.org/10.1007/978-3-0348-9210-0_1

P. E. Crouch. Solvable approximations to control systems, SIAM J. Control Optimiz. -- 1984. -- Vol. 22. -- P. 40--54. DOI: http://doi.org/10.1137/0322004

M. Fliess. Realization of nonlinear systems and abstract transitive Lie algebras, Bull. of the AMS. -- 1980. -- Vol. 2. -- P.~444--446. DOI: http://doi.org/10.1090/S0273-0979-1980-14760-6

M. Fliess. Fonctionnelles causales non lin'{e}aires et ind'{e}termin'{e}es non commutatives, Bull. Soc. Math. France. -- 1981. -- Vol. 109. -- P.~3--40.

H. Hermes. Nilpotent and high-order approximations of vector field systems, SIAM Rev. -- 1991. -- Vol. 33. -- P. 238--264. DOI: http://doi.org/10.1137/1033050

S. Yu. Ignatovich. Realizable growth vectors of affine control systems, J. Dyn. Control Syst. -- 2009. -- Vol. 15. -- P.~557--585. DOI: http://doi.org/10.1007/s10883-009-9075-y

A. Isidori. Nonlinear control systems. 3-rd ed. Springer-Verlag, London. -- 1995. -- 549 p. DOI: https://doi.org/10.1007/978-1-84628-615-5

B. Jakubczyk. Existence and uniqueness of realizations of nonlinear systems, SIAM J. Control and Optimiz. -- 1980. -- Vol. 18. -- P.~455--471. DOI: http://doi.org/10.1137/0318034

F. Jean. Control of nonholonomic systems: from sub-Riemannian geometry to motion planning, Springer Cham. -- 2014. -- 104 p. DOI: https://doi.org/10.1007/978-3-319-08690-3

M. Kawski. Nonlinear control and combinatorics of words, in: Geometry of Feedback and Optimal Control, Dekker. -- 1997. -- P. 305--346.

M. Kawski. Combinatorial algebra in controllability and optimal control, in: Algebra and Applications 2: Combinatorial Algebra and Hopf Algebras, A. Makhlouf, ed., Chapter 5. -- 2021. -- P.~221--286. DOI: https://doi.org/10.1002/9781119880912.ch5

M. Kawski, H. J. Sussmann. Noncommutative power series and formal Lie-algebraic techniques in nonlinear control theory, in: Operators, Systems and Linear Algebra. European Consortium for Mathematics in Industry, U.~Helmke, D. Pr"{a}tzel-Wolters, E. Zerz, eds., Teubner. -- 1997. -- P.~111--128. DOI: http://doi.org/10.1007/978-3-663-09823-2_10

G. Melanc{c}on, C. Reutenauer. Lyndon words, free algebras and shuffles, Canad. J. Math. -- 1989. -- Vol. textbf{41}. -- P.~577--591. DOI: http://doi.org/10.4153/CJM-1989-025-2

P. Mormul, F. Pelletier. Symmetries of special 2-flags, Journal of Singularities. -- 2020. -- Vol. textbf{21}. -- P.~187--204. DOI: http://doi.org/10.5427/jsing.2020.21k

C. Reutenauer. Free Lie algebras. Clarendon Press, Oxford. -- 1993. -- 286 p.

G. M. Sklyar, S. Yu. Ignatovich. Moment approach to nonlinear time optimality, SIAM J. Control Optimiz. -- 2000. -- Vol. 38. -- P.~1707--1728. DOI: http://doi.org/10.1137/S0363012997329767

G. M. Sklyar, S. Yu. Ignatovich. Approximation of time-optimal control problems via nonlinear power moment min-problems, SIAM J. Control Optimiz. -- 2003. -- Vol. textbf{42}. -- P.~1325--1346. DOI: http://doi.org/10.1137/S0363012901398253

G. M. Sklyar, S. Yu. Ignatovich. Free algebras and noncommutative power series in the analysis of nonlinear control systems: an application to approximation problems, Dissertationes Math. (Rozprawy Mat.) -- 2014. -- Vol. 504. -- P.~1--88. DOI: http://doi.org/10.4064/dm504-0-1

G. Sklyar, P. Barkhayev, S. Ignatovich, V. Rusakov. Implementation of the algorithm for constructing homogeneous approximations of nonlinear control systems, Mathematics of Control, Signals, and Systems. -- 2022. -- Vol. 34. -- P.~883--907. DOI: http://doi.org/10.1007/s00498-022-00330-5

G. Stefani. Polynomial approximations to control systems and local controllability, in: 1985 24th IEEE Conference on Decision and Control. -- 1985. -- P.~33--38. DOI: http://doi.org/10.1109/CDC.1985.268467

Published
2022-12-24
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How to Cite
Andreieva, D. M., & Ignatovich, S. Y. (2022). Homogeneous approximation for minimal realizations of series of iterated integrals. Visnyk of V. N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics, 96, 23-39. https://doi.org/10.26565/2221-5646-2022-96-02
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