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Volumn 52, Issue 12, 2007, Pages 2227-2241

From chronological calculus to exponential representations of continuous and discrete-time dynamics: A Lie-algebraic approach

Author keywords

Chronological calculus; Discrete time and sampled dynamics; Nonautonomous differential equations; Nonlinear systems

Indexed keywords

ALGEBRA; CONTROL THEORY; DIFFERENTIAL EQUATIONS; DISCRETE TIME CONTROL SYSTEMS;

EID: 38149128006     PISSN: 00189286     EISSN: None     Source Type: Journal    
DOI: 10.1109/TAC.2007.902734     Document Type: Article
Times cited : (46)

References (38)
  • 1
    • 84940252991 scopus 로고
    • On the exponential solution of differential equations for a linear operator
    • W. Magnus, "On the exponential solution of differential equations for a linear operator," Commun. Pure and Appl. Math., vol. 7, pp. 649-673, 1954.
    • (1954) Commun. Pure and Appl. Math , vol.7 , pp. 649-673
    • Magnus, W.1
  • 2
    • 0242595298 scopus 로고
    • Formal differential equations
    • K. T. Chen, "Formal differential equations," Annals of Math., vol. 73, pp. 110-133, 1961.
    • (1961) Annals of Math , vol.73 , pp. 110-133
    • Chen, K.T.1
  • 3
    • 0008328552 scopus 로고
    • Serie di Lie e Loro Applicazioni
    • Cremonese, Italy
    • W. Gröbner, "Serie di Lie e Loro Applicazioni," in Poliedro, Cremonese, Italy, 1973.
    • (1973) Poliedro
    • Gröbner, W.1
  • 4
    • 84956258691 scopus 로고
    • Exponential representations of flows and chronological calculus
    • A. A. Agrachev and R. V. Gamkrelidze, "Exponential representations of flows and chronological calculus," Math. USSR Sbornik, vol. 35, pp. 727-785, 1978.
    • (1978) Math. USSR Sbornik , vol.35 , pp. 727-785
    • Agrachev, A.A.1    Gamkrelidze, R.V.2
  • 5
    • 0001843817 scopus 로고
    • Fonctionnelles causales non linéaires et indéterminées non commutatives
    • M. Fliess, "Fonctionnelles causales non linéaires et indéterminées non commutatives," Bull. Soc. Math. France, vol. 109, pp. 3-40, 1981.
    • (1981) Bull. Soc. Math. France , vol.109 , pp. 3-40
    • Fliess, M.1
  • 6
    • 0021584553 scopus 로고
    • A Lie Volterra expansion for nonlinear systems
    • H. J. Sussmann, "A Lie Volterra expansion for nonlinear systems," Lecture Notes Cont. Inf. Sci., vol. 58, pp. 822-828, 1984.
    • (1984) Lecture Notes Cont. Inf. Sci , vol.58 , pp. 822-828
    • Sussmann, H.J.1
  • 7
    • 0000011103 scopus 로고
    • Volterra series and geometric control
    • R. W. Brockett, "Volterra series and geometric control," Automatica vol. 12, pp. 167-176, 1976.
    • (1976) Automatica , vol.12 , pp. 167-176
    • Brockett, R.W.1
  • 8
    • 0018152524 scopus 로고
    • The existence and uniqueness of Volterra series for nonlinear systems
    • K. Lesiak and A. J. Krener, "The existence and uniqueness of Volterra series for nonlinear systems," IEEE Trans. Autom. Control, vol. 23, pp. 1090-1095, 1978.
    • (1978) IEEE Trans. Autom. Control , vol.23 , pp. 1090-1095
    • Lesiak, K.1    Krener, A.J.2
  • 10
    • 0020708986 scopus 로고
    • Volterra-like expansions for solutions of nonlinear integral equations and nonlinear differential equations
    • I. W. Sandberg, "Volterra-like expansions for solutions of nonlinear integral equations and nonlinear differential equations," IEEE Trans. Autom. Control, vol. 30, pp. 68-77, 1983.
    • (1983) IEEE Trans. Autom. Control , vol.30 , pp. 68-77
    • Sandberg, I.W.1
  • 12
    • 0023027733 scopus 로고
    • A product expansion of the Chen series
    • C. I. Byrnes and A. Lindquist, Eds. North Holland, Elsevier
    • H. J. Sussmann, "A product expansion of the Chen series," in Theory and Applications of Nonlinear Control Systems, C. I. Byrnes and A. Lindquist, Eds. North Holland,: Elsevier, 1986, pp. 323-335.
    • (1986) Theory and Applications of Nonlinear Control Systems , pp. 323-335
    • Sussmann, H.J.1
  • 13
    • 0002038341 scopus 로고    scopus 로고
    • Noncommutative power series and formal Lie algebraic techniques in control theory
    • U. Helmke, D. P. Wolters, and E. Zertz, Eds. Studgart, Germany: Springer
    • M. Kawski and H. J. Sussmann, "Noncommutative power series and formal Lie algebraic techniques in control theory," in Systems and Linear Algebra, U. Helmke, D. P. Wolters, and E. Zertz, Eds. Studgart, Germany: Springer, 1997, pp. 111-128.
    • (1997) Systems and Linear Algebra , pp. 111-128
    • Kawski, M.1    Sussmann, H.J.2
  • 15
    • 0042419252 scopus 로고
    • Algèbres de Lie nilpotentes, formule de Baker-Campbell-Hausdorff et intégrales itérées de K.T. Chen
    • Séminaire de Probabilités, J. Azema and M. Yor, Eds. New York: Springer-Verlag
    • M. Fliess and D. Normand-Cyrot, "Algèbres de Lie nilpotentes, formule de Baker-Campbell-Hausdorff et intégrales itérées de K.T. Chen," in Séminaire de Probabilités, J. Azema and M. Yor, Eds. New York: Springer-Verlag, 1980-1981, vol. 16, pp. 257-267, Lecture Notes Math.
    • (1980) Lecture Notes Math , vol.16 , pp. 257-267
    • Fliess, M.1    Normand-Cyrot, D.2
  • 16
    • 85083151587 scopus 로고    scopus 로고
    • B. Jakubczyk and D. Normand-Cyrot, Orbites de pseudo groupes de difféomorphismes et commandabilité des systèmes non linéaires en temps discret, in C.R. Acad. Sci., Paris, France, 1984, 298-I-11, pp. 257-260.
    • B. Jakubczyk and D. Normand-Cyrot, "Orbites de pseudo groupes de difféomorphismes et commandabilité des systèmes non linéaires en temps discret," in C.R. Acad. Sci., Paris, France, 1984, vol. 298-I-11, pp. 257-260.
  • 17
    • 0025249659 scopus 로고
    • Controllability of nonlinear discrete time systems: A Lie algebraic approach
    • B. Jakubczyk and E. D. Sontag, "Controllability of nonlinear discrete time systems: A Lie algebraic approach," SIAM J. Contr. Optim., vol. 28, pp. 1-33, 1990.
    • (1990) SIAM J. Contr. Optim , vol.28 , pp. 1-33
    • Jakubczyk, B.1    Sontag, E.D.2
  • 18
    • 0031706909 scopus 로고    scopus 로고
    • Chronological calculus, high-order necessary conditions for optimality, and perturbation methods
    • A. I. Tretyak, "Chronological calculus, high-order necessary conditions for optimality, and perturbation methods," J. Dyn. Control Syst., vol. 4, pp. 77-126, 1998.
    • (1998) J. Dyn. Control Syst , vol.4 , pp. 77-126
    • Tretyak, A.I.1
  • 19
    • 0033333363 scopus 로고    scopus 로고
    • Controllability via chronological calculus
    • M. Kawski, "Controllability via chronological calculus," Proc. IEEE-CDC, vol. 3, pp. 2920-25, 1999.
    • (1999) Proc. IEEE-CDC , vol.3 , pp. 2920-2925
    • Kawski, M.1
  • 20
    • 84949087689 scopus 로고    scopus 로고
    • Geometric properties of a class of nonlinear discrete-time dynamics
    • Brussels, Belgium
    • S. Monaco and D. Normand-Cyrot, "Geometric properties of a class of nonlinear discrete-time dynamics," in Proc. ECC-97, Brussels, Belgium, 1997.
    • (1997) Proc. ECC-97
    • Monaco, S.1    Normand-Cyrot, D.2
  • 21
    • 2442472285 scopus 로고
    • Exponential representations solutions of ordinary differential equations
    • R. V. Gamkrelidze, "Exponential representations solutions of ordinary differential equations," Lecture Notes Math., vol. 703, 1979.
    • (1979) Lecture Notes Math , vol.703
    • Gamkrelidze, R.V.1
  • 22
    • 0034911587 scopus 로고    scopus 로고
    • Lie- and chronologico-algebraic tools for studying stability of time-varying systems
    • A. V. Sarychev, "Lie- and chronologico-algebraic tools for studying stability of time-varying systems," Syst. Control Lett., vol. 43, pp. 59-76, 2001.
    • (2001) Syst. Control Lett , vol.43 , pp. 59-76
    • Sarychev, A.V.1
  • 23
    • 1542379849 scopus 로고    scopus 로고
    • Discrete-time versus hybrid systems
    • Maui, HI
    • S. Monaco, D. Normand-Cyrot, and C. Califano, "Discrete-time versus hybrid systems," in Proc. IEEE-CDC-03, Maui, HI, 2003, vol. 5, pp. 5203-5208.
    • (2003) Proc. IEEE-CDC-03 , vol.5 , pp. 5203-5208
    • Monaco, S.1    Normand-Cyrot, D.2    Califano, C.3
  • 24
    • 0033877553 scopus 로고    scopus 로고
    • A hybrid control scheme for maneuvering space multibody systems
    • P. Di Giamberardino, S. Monaco, and D. Normand-Cyrot, "A hybrid control scheme for maneuvering space multibody systems," J. Guid. Control Dyn., vol. 23, no. 2, pp. 231-240, 2000.
    • (2000) J. Guid. Control Dyn , vol.23 , Issue.2 , pp. 231-240
    • Di Giamberardino, P.1    Monaco, S.2    Normand-Cyrot, D.3
  • 25
    • 0030648872 scopus 로고
    • A unifying representation for nonlinear discrete-time and sampled dynamics
    • summary
    • S. Monaco and D. Normand-Cyrot, "A unifying representation for nonlinear discrete-time and sampled dynamics," J. Math. Syst., Est. Cont., vol. 7, pp. 477-503, 1997, (1995), pp. 101-103 (summary).
    • (1995) J. Math. Syst., Est. Cont , vol.7
    • Monaco, S.1    Normand-Cyrot, D.2
  • 26
    • 8744267214 scopus 로고    scopus 로고
    • Issues on nonlinear digital systems
    • S. Monaco and D. Normand-Cyrot, "Issues on nonlinear digital systems," Europ. J. Contr., vol. 7, pp. 160-178, 2001.
    • (2001) Europ. J. Contr , vol.7 , pp. 160-178
    • Monaco, S.1    Normand-Cyrot, D.2
  • 28
    • 8744260769 scopus 로고    scopus 로고
    • Exponential representations of multi-input nonlinear discrete-time dynamics
    • Boston, MA
    • S. Monaco, D. Normand-Cyrot, and C. Califano, "Exponential representations of multi-input nonlinear discrete-time dynamics," in Proc. ACC-04, Boston, MA, 2004, pp. 4998-5003.
    • (2004) Proc. ACC-04 , pp. 4998-5003
    • Monaco, S.1    Normand-Cyrot, D.2    Califano, C.3
  • 29
    • 0037161312 scopus 로고    scopus 로고
    • A combinatorial approach to the generalized Baker Campbell Hausdorff Dynkin formula
    • L. Saenz and R. Suarez, "A combinatorial approach to the generalized Baker Campbell Hausdorff Dynkin formula," Syst. Control Lett., vol. 45, pp. 357-370, 2002.
    • (2002) Syst. Control Lett , vol.45 , pp. 357-370
    • Saenz, L.1    Suarez, R.2
  • 30
    • 0000483646 scopus 로고
    • Lie elements and an algebra associated with shuffles
    • R. Ree, "Lie elements and an algebra associated with shuffles," Ann. Math., vol. 68, pp. 210-220, 1958.
    • (1958) Ann. Math , vol.68 , pp. 210-220
    • Ree, R.1
  • 31
    • 0000389923 scopus 로고    scopus 로고
    • Stability criteria for time-periodic systems via highorder averaging techniques
    • Berlin, Germany: Springer-Verlag
    • A. V. Sarychev, "Stability criteria for time-periodic systems via highorder averaging techniques," in Nonlinear Control in the Year 2000, Lecture Notes in Contr. Inf. Sci.. Berlin, Germany: Springer-Verlag, 2001, vol. 259-2, pp. 365-377.
    • (2001) Nonlinear Control in the Year 2000, Lecture Notes in Contr. Inf. Sci , vol.259 -2 , pp. 365-377
    • Sarychev, A.V.1
  • 35
    • 0024068815 scopus 로고
    • Feedback linearization of sampled-data systems
    • J. W. Grizzle and P. V.Kokotovic, "Feedback linearization of sampled-data systems," IEEE Trans. Autom. Control, vol. 33, pp. 857-859, 1988.
    • (1988) IEEE Trans. Autom. Control , vol.33 , pp. 857-859
    • Grizzle, J.W.1    Kokotovic, P.V.2
  • 36
    • 3843084016 scopus 로고    scopus 로고
    • A framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time models
    • D. Nesic and A. R. Teel, "A framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time models," IEEE Trans. Autom. Control, vol. 49, pp. 1103-1034, 2004.
    • (2004) IEEE Trans. Autom. Control , vol.49 , pp. 1103-1034
    • Nesic, D.1    Teel, A.R.2
  • 37
    • 0023346788 scopus 로고
    • Finite Volterra series realizations and input output approximations of nonlinear discrete-time systems
    • S. Monaco and D. Normand-Cyrot, "Finite Volterra series realizations and input output approximations of nonlinear discrete-time systems," Int. J. Control, vol. 45-5, pp. 1771-1787, 1987.
    • (1987) Int. J. Control , vol.45 -5 , pp. 1771-1787
    • Monaco, S.1    Normand-Cyrot, D.2
  • 38
    • 3342940326 scopus 로고
    • Functional expansions for non-linear discrete-time systems
    • S. Monaco and D. Normand-Cyrot, "Functional expansions for non-linear discrete-time systems," Math. Syst. Theory, vol. 21, pp. 235-254, 1989.
    • (1989) Math. Syst. Theory , vol.21 , pp. 235-254
    • Monaco, S.1    Normand-Cyrot, D.2


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