Abstract
In this paper we continue the analysis of the problem of output tracking in the presence of singularities, whose study was begun by R. Hirschorn and J. Davis. We introduce further structure which is important in quantifying the multiplicity and smoothness of solutions to the problem. The paper is motivated by the analysis of those singular ordinary differential equations whose structure ultimately governs solutions to the singular tracking problem. In the particular case of time-varying linear systems, it is shown how the structure of their solutions in the case of regular and irregular singularities affects solutions to the tracking problem. Less specific results are also obtained in the full nonlinear case.
Similar content being viewed by others
References
[C] P. E. Crouch, Graded vector spaces and applications to the approximation of nonlinear systems,Rendiconti Del Seminario Matematico: Linear and Nonlinear Mathematical Control Theory, Torino, 1987, pp. 1–54.
[CL] P. E. Crouch and F. Lamnabhi-Lagarrigue, State space realizations of nonlinear systems defined by input-output differential equations, inAnalysis and Optimization of Systems (A. Bensoussan and J. L. Lions, eds.), pp. 138–149, Lecture Notes in Control and Information Science, Vol. 111, Springer-Verlag, Berlin, 1988.
[E] A. Erdelyi,Asymptotic Expansions, Dover, New York, 1956.
[F1] M. Fliess, A note on the invertibility of nonlinear input-output differential equations.Systems Control Lett.,8 (1986), 147–151.
[F2] M. Fliess, Nonlinear control theory and differential algebra, inModeling and Adaptative Control (C. I. Byrnes and A. B. Kurzhanski, eds.), pp. 134–145, Lecture Notes in Control and Information Sciences, Vol. 105, Springer-Verlag, Berlin, 1988.
[G1] S. T. Glad, Nonlinear state space and input-output descriptions using differential polynomials, inNew Trends in Nonlinear Control Theory (J. Descusse, M. Fliess, A. Isidori, and D. Leborgne, eds.), pp. 182–189, Lecture Notes in Control and Information Sciences, Vol. 122, Springer-Verlag, Berlin, 1989.
[G2] K. A. Grasse, Sufficient conditions for the functional reproducibility of time-varying input-output systems,SIAM J. Control Optim.,26 (1988), 230–249.
[H1] E. Hille,Ordinary Differential Equations in the Complex Domain, Wiley, New York, 1976.
[H2] R. M. Hirschorn, Invertibility of nonlinear control systems,SIAM J. Control Optim.,17 (1979), 289–297.
[H3] R. M. Hirschorn, Output tracking into multivariable nonlinear systems,IEEE Trans. Automat. Control,26 (1981), 593–595.
[HD1] R. M. Hirschorn and J. Davis, Output tracking for nonlinear systems with singular points,SIAM J. Control Optim.,26 (1987), 547–557.
[HD2] R. M. Hirschorn and J. Davis, Global output tracking for nonlinear systems,SIAM J. Control Optim.,27 (1988), 1321–1330.
[I] E. L. Ince,Ordinary Differential Equations, Dover, New York, 1956.
[IM] A. Isidori and C. H. Moog, On the nonlinear equivalent of the notion of transmission zeros, inModeling and Adaptive Control (C. I. Byrnes and A. B. Kurzhanski, eds.), Lecture Notes in Control and Information Sciences, Vol. 105, Springer-Verlag, Berlin, 1988.
[K] A. J. Krener, On the equivalence of control systems and the linearization of nonlinear systems,SIAM J. Control Optim.,11 (1973), 670–676.
[LCI] F. Lamnabhi-Lagarrigue, P. E. Crouch, and I. Ighneiwa, Output tracking through singularities, inPerspectives in Control Theory (B. Jakubczyk, K. Malanauski and W. Respondek, eds.), pp. 154–174, Birkhäuser Boston, Cambridge, MA, 1990.
[LB] I. J. Leontaritis and S. A. Billings, Input-output parametric models for nonlinear systems, Parts I and II,Internat. J. Control,41 (1985), 303–344.
[N] H. Neimeijer, Invertibility of affine nonlinear control systems: a geometric approach.Systems Control Lett.,2 (1982), 163–168.
[S1] L. M. Silverman, Properties and applications of inverse systems,IEEE Trans. Automat. Control,13 (1968), 436–438.
[S2] S. N. Singh, Reproductibility in nonlinear systems using dynamic compensation and output feedback,IEEE Trans. Automat. Control,27 (1982), 955–925.
[S3] S. N. Singh, Generalized functional reproducibility condition for nonlinear systems,IEEE Trans. Automat. Control,27 (1982), 958–960.
[S4] E. D. Sontag,Polynomial Response Maps, Lecture Notes in Control and Information Sciences, Vol. 13, Springer-Verlag, Berlin, 1979.
[S5] E. D. Sontag, Bilinear realizability is equivalent to existence of a singular affine differential I/O equation,System Control Lett.,11 (1988), 181–187.
[V1] A. van der Schaft, On realization of nonlinear systems described by higher-order differential equations,Math. Systems Theory,19 (1987), 239–275.
[V2] A. van der Schaft, Representing a nonlinear state space system as a set of higher-order differential equations in the inputs and outputs,Systems Control Lett.,12 (1989), 151–160.
[V3] A. van der Schaft, Transformations of nonlinear-system under external equivalence, inNew Trends in Nonlinear Control Theory (J. Descusse, M. Fliess, A. Isidori, and D. Leborgne, eds.), pp. 33–43, Lecture Notes in Control and Information Sciences, Vol. 122, Springer-Verlag, Berlin, 1989.
[WS1] Y. Wang, and E. D. Sontag, On two definitions of observation spaces,Systems Control Lett.,13 (1989), 279–299.
[WS2] Y. Wang and E. D. Sontag, Realization and input/output relations: the analytic case,Proceedings of the 28th IEEE Conference on Decision and Control, Tampa, FL, 1989, pp. 1975–1980.
[WS3] Y. Wang and E. D. Sontag, Input/output equations and realizability, inRealization and Modelling in System Theory (M. A. Kaashoek, J. H. van Schuppen, and A. C. M. Ran, eds.), pp. 125–132, Birkhäuser Boston, Cambridge, MA, 1990.
[W] W. Wasow,Asymptotic Expansions for Ordinary Differential Equations, Pure and Applied Mathematics, Vol. 14, Wiley Interscience, New York, 1965.
[X] Xiaoming Hu, Feedback stabilization and tracking for nonlinear systems, Ph.D. Thesis, Arizona State University, Phoenix, Arizona, 1989.
[Z] M. Zeitz, Observability canonical (phase-variable) form for nonlinear time variable systems,Internat. J. Systems Science,15 (1984), 949–958.
Author information
Authors and Affiliations
Additional information
P. E. Crouch and I. Ighneiwa were partially supported by N.S.F. Contract No. ECS 8703615.
Rights and permissions
About this article
Cite this article
Crouch, P.E., Ighneiwa, I. & Lamnabhi-Lagarrigue, F. On the singular tracking problem. Math. Control Signal Systems 4, 341–362 (1991). https://doi.org/10.1007/BF02570567
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02570567