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Free algebras and noncommutative power series in the analysis of nonlinear control systems: an application to approximation problems. (English) Zbl 1311.93019
Summary: The paper contains a consistent presentation of the approach developed by the authors to analysis of nonlinear control systems, which exploits ideas and techniques of formal power series of independent noncommuting variables and the corresponding free algebras. The main part of the paper is conceived with a view of comparing our results with the results obtained by use of the differential-geometric approach. We consider control-linear systems with $$m$$ controls. In a free associative algebra with $$m$$ generators (which can be thought of as a free algebra of iterated integrals), a control system uniquely defines two special objects: the core Lie subalgebra and the graded left ideal. It turns out that each of these two objects completely defines a homogeneous approximation of the system. Our approach allows us to propose an algebraic (coordinate-independent) definition of the homogeneous approximation. This definition provides the uniqueness of the homogeneous approximation (up to a change of coordinates) and gives a way to find it directly, without preliminary finding privileged coordinates. The presented technique yields an effective description of all privileged coordinates and an explicit way of constructing an approximating system. In addition, we discuss the connection between the homogeneous approximation and an approximation in the sense of time optimality.

##### MSC:
 93B25 Algebraic methods 93B11 System structure simplification 93B15 Realizations from input-output data 93B17 Transformations
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 [1] [1]A. A. Agrachev and R. V. Gamkrelidze, Exponential representation of flows and a chronological enumeration, Mat. Sb. (N.S.) 107 (1978), no. 4, 467–532 (in Russian); English transl.: Math. USSR-Sb. 35 (1979), 727–785. · Zbl 0408.34044 [2] [2]A. A. Agrachev and R. V. Gamkrelidze, The shuffle product and symmetric groups, in: Differential Equations, Dynamical Systems, and Control Science, Lecture Notes Pure Appl. Math. 152, Dekker, New York, 1998, 365–382. · Zbl 0790.05095 [3] [3]A. A. Agrachev, R. V. Gamkrelidze, and A. V. Sarychev, Local invariants of smooth control systems, Acta Appl. Math. 14 (1989), 191–237. · Zbl 0681.49018 [4] [4]A. Agrachev and A. Marigo, Nonholonomic tangent spaces: intrinsic construction and rigid dimensions, Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 111–120. · Zbl 1068.58001 [5] [5]N. I. Akhiezer and M. G. Krein, Some Questions in the Theory of Moments, GONTI, Kharkiv, 1938 (in Russian); English transl.: Transl. Math. Monogr. 2, Amer. Math. Soc., Providence, RI, 1962. [6] [6]A. Bella\"{}ıche, The tangent space in sub-Riemannian geometry, in: Sub-Riemannian Geometry, Progr. Math. 144, Birkh\"{}auser, Basel, 1996, 4–78. · Zbl 0862.53031 [7] [7]A. Bella\"{}ıche, F. Jean, and J.-J. Risler, Geometry of nonholonomic systems, in: J.-P. Laumond (ed.), Robot Motion Planning and Control, Lecture Notes in Control and Inform. Sci. 229, Springer, 1998, 55–92. [8] [8]R. M. Bianchini and G. Stefani, Graded approximation and controllability along a trajectory, SIAM J. Control Optim. 28 (1990), 903–924. · Zbl 0712.93005 [9] [9]R. W. Brockett, Volterra series and geometric control theory, Automatica J. IFAC 12 (1976), 167–176. · Zbl 0342.93027 [10] [10]K. T. Chen, Integration of paths–a faithful representation of parths by noncommutative formal power series, Trans. Amer. Math. Soc. 89 (1958), 395–407. [11] [11]W. L. Chow, \"{}Uber Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann. 117 (1939), 98–105. · JFM 65.0398.01 [12] [12]P. E. Crouch, Solvable approximations to control systems, SIAM J. Control Optim. 22 (1984), 40–54. · Zbl 0537.93040 [13] [13]P. E. Crouch and F. Lamnabhi-Lagarrigue, Algebraic and multiple integral identities, Acta Appl. Math. 15 (1989), 235–274. · Zbl 0677.93041 [14] [14]S. Eilenberg and S. Mac Lane, On the groups H({$$\Pi$$}, n), I, Ann. of Math. 58 (1953), 55–106. [15] [15]A. F. Filippov, On some questions in the theory of optimal regulation: existence of a solution of the problem of optimal regulation in the class of bounded measurable functions, Vestnik Moskov. Univ. Ser. Mat. Meh. Astr. Fiz. Him. 1959, no. 2, 25–32 (in Russian). [16] [16]A. F. Filippov, On certain questions in the theory of optimal control, J. SIAM Control Ser. A 1 (1962), 76–84. · Zbl 0139.05102 [17] [17]M. Fliess, D\'{}eveloppements fonctionnels en ind\'{}etermin\'{}ees non commutatives des solutions d’\'{}equations diff\'{}erentielles non lin\'{}eaires forc\'{}ees, C. R. Acad. Sci. Paris S\'{}er. A–B 287 (1978), 1133–1135. · Zbl 0402.93027 [18] [18]M. Fliess, Realizations of nonlinear systems and abstract transitive Lie algebras, Bull. Amer. Math. Soc. 2 (1980), 444–446. · Zbl 0427.93011 [19] [19]M. Fliess, Fonctionnelles causales non lin\'{}eaires et ind\'{}etermin\'{}ees non commutatives, Bull. Soc. Math. France 109 (1981), 3–40. · Zbl 0476.93021 [20] [20]M. Fliess, M. Lamnabhi, and F. Lamnabhi-Lagarrigue, An algebraic approach to nonlinear functional expansions, IEEE Trans. Circuits and Systems 30 (1983), 554–570. Free algebras and noncommutative power series in nonlinear control problems87 · Zbl 0529.34002 [21] [21]E. Gehrig and M. Kawski, A Hopf-algebraic formula for compositions of noncommuting flows, in: Proc. 47th IEEE Conference on Decision and Control, 2008, 1569–1574. [22] [22]E. G. Gilbert, Functional expansions for the response of nonlinear differential systems, IEEE Trans. Automatic Control AC-22 (1977), 909–921. · Zbl 0373.93022 [23] [23]H. Hermes, Nilpotent approximations of control systems and distributions, SIAM J. Control Optim. 24 (1986), 731–736. · Zbl 0604.93031 [24] [24]H. Hermes, Nilpotent and high-order approximations of vector field systems, SIAM Rev. 33 (1991), 238–264. · Zbl 0733.93062 [25] [25]S. Yu. Ignatovich, Realizable growth vectors of affine control systems, J. Dynam. Control Systems 15 (2009), 557–585. · Zbl 1203.93098 [26] [26]S. Yu. Ignatovich, Normalization of homogeneous approximations of symmetric affine control systems with two controls, J. Dynam. Control Systems 17 (2011), 1–48. · Zbl 1211.93030 [27] [27]A. Isidori, Nonlinear control systems: an introduction, in: Lecture Notes in Control and Inform. Sci. 72, Springer, Berlin, 1985. · Zbl 0569.93034 [28] [28]B. Jakubczyk, Local realizations of nonlinear causal operators, SIAM J. Control Optim. 24 (1986), 230–242. · Zbl 0613.93010 [29] [29]B. Jakubczyk, Realization theory for nonlinear systems; three approaches, in: M. Fliess and M. Hazewinkel (eds.), Algebraic and Geometric Methods in Nonlinear Control Theory, Reidel, Dordrecht, 1986, 3–31. · Zbl 0608.93018 [30] [30]B. Jakubczyk, Convergence of power series along vector fields and their commutators; a Cartan–K\"{}ahler type theorem, Ann. Polon. Math. 74 (2000), 117–132. · Zbl 0961.35023 [31] [31]F. Jean, Uniform estimation of sub-Riemannian balls, J. Dynam. Control Systems 7 (2001), 473–500. · Zbl 1029.53039 [32] [32]M. Kawski, Nonlinear control and combinatorics of words, in: Geometry of Feedback and Optimal Control, Monogr. Textbooks Pure Appl. Math. 207, Dekker, New York, 1998, 305–346. · Zbl 0925.93368 [33] [33]M. Kawski, The combinatorics of nonlinear controllability and noncommuting flows, in: ICTP Lect. Notes 8, Trieste, 2002, 223–312. · Zbl 1098.93500 [34] [34]M. Kawski, Control interpretations of products in the Hopf algebra, in: Proc. 48th IEEE Conference on Decision and Control, 2009, 7503–7508. [35] [35]M. Kawski and H. Sussmann, Noncommutative power series and formal Lie-algebraic techniques in nonlinear control theory, in: Operators, Systems, and Linear Algebra, Teubner, 1997, 111–128. · Zbl 0919.93035 [36] [36]V. I. Korobov, The continuous dependence of a solution of an optimal-control problem with a free time for initial data, Differentsial’nye Uravneniya 7 (1971), 1120–1123 (in Russian); English transl.: Differ. Equations 7 (1971), 850–852 (1973). · Zbl 0224.49007 [37] [37]V. I. Korobov and G. M. Sklyar, Time-optimality and the power moment problem, Mat. Sb. (N.S.) 134 (176) (1987), no. 2, 186–206, 287 (in Russian); English transl.: Math. USSR-Sb. 62 (1989), no. 1, 185–206. · Zbl 0639.93034 [38] [38]V. I. Korobov and G. M. Sklyar, The Markov moment min-problem and time optimality, Sibirsk. Mat. Zh. 32 (1991), no. 1, 60–71, 220 (in Russian); English transl.: Siberian Math. J. 32 (1991), no. 1, 46–55. [39] [39]M. G. Krein and A. A. Nudel’man, The Markov moment problem and extremal problems. Ideas and problems of P. L. Chebyshev and A. A. Markov and their further development, Nauka, Moscow, 1973 (in Russian); English transl.: Transl. Math. Monogr. 50, Amer. Math. Soc., Providence, RI, 1977. 88G. M. Sklyar and S. Yu. Ignatovich [40] [40]C. Lesiak and A. Krener, The existence and uniqueness of Volterra series for nonlinear systems, IEEE Trans. Automat. Control 23 (1978), 1090–1095. · Zbl 0393.93009 [41] [41]W. S. Liu and H. J. Sussmann, Shortest paths for sub-Riemannian metrics of rank two distributions, Mem. Amer. Math. Soc. 118 (1995), no. 564. [42] [42]M. Lothaire, Combinatorics on Words, Addison-Wesley, Reading, MA, 1983. [43] [43]A. A. Markov, Nouvelles applications des fractions continues, Math. Ann. 47 (1896), 579– 597. · JFM 27.0176.01 [44] [44]G. Melan\c{}con and C. Reutenauer, Lyndon words, free algebras and shuffles, Canad. J. Math. 41 (1989), 577–591. · Zbl 0694.17003 [45] [45]P. K. Rashevski, About connecting two points of a completely nonholonomic space by an admissible curve, Uch. Zapiski Ped. Inst. Liebknechta 2 (1938), 83–94 (in Russian). [46] [46]R. Ree, Lie elements and an algebra associated with shuffles, Ann. of Math. 68 (1958), 210–220. · Zbl 0083.25401 [47] [47]C. Reutenauer, Free Lie Algebras, Clarendon Press, Oxford, 1993. [48] [48]G. M. Sklyar and S. Yu. Ignatovich, A classification of linear time-optimal control problems in a neighborhood of the origin, J. Math. Anal. Appl. 203 (1996), 791–811. · Zbl 0877.49003 [49] [49]G. M. Sklyar and S. Yu. Ignatovich, Moment approach to nonlinear time optimality, SIAM J. Control Optim. 38 (2000), 1707–1728. · Zbl 0989.93093 [50] [50]G. M. Sklyar and S. Yu. Ignatovich, Representations of control systems in the Fliess algebra and in the algebra of nonlinear power moments, Systems Control Lett. 47 (2002), 227–235. · Zbl 1106.93309 [51] [51]G. M. Sklyar and S. Yu. Ignatovich, Approximation of time-optimal control problems via nonlinear power moment min-problems, SIAM J. Control Optim. 42 (2003), 1325–1346. · Zbl 1049.93010 [52] [52]G. M. Sklyar and S. Yu. Ignatovich, Determining of various asymptotics of solutions of nonlinear time optimal problems via right ideals in the moment algebra (Problem 3.8), in: V. D. Blondel and A. Megretski (eds.), Unsolved Problems in Mathematical Systems and Control Theory, Princeton Univ. Press, Princeton, 2004, 117–121. [53] [53]G. M. Sklyar and S. Yu. Ignatovich, Description of all privileged coordinates in the homogeneous approximation problem for nonlinear control systems, C. R. Math. Acad. Sci. Paris 344 (2007), 109–114. · Zbl 1134.93013 [54] [54]G. M. Sklyar and S. Yu. Ignatovich, Fliess series, a generalization of the Ree’s theorem, and an algebraic approach to a homogeneous approximation problem, Int. J. Control 81 (2008), 369–378. · Zbl 1152.93334 [55] [55]G. M. Sklyar, S. Yu. Ignatovich, and P. Yu. Barkhaev, Algebraic classification of nonlinear steering problems with constraints on control, in: G. Oyibo (ed.), Adv. Math. Res. 6, Nova Science, 2005, 37–96. [56] [56]H. J. Sussmann, A product expansion for the Chen series, in: Theory and Applications of Nonlinear Control Systems (Stockholm, 1985), North-Holland, Amsterdam, 1986, 323–335. [57] [57]Y. Wang and E. D. Sontag, Generating series and nonlinear systems: analytic aspects, local realizability, and i/o representations, Forum Math. 4 (1992), 299–322. · Zbl 0746.93020
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