Skip to main content
SpringerLink
Log in

An Algebraic Approach to Linear and Nonlinear Control

  • Chapter
Essays on Control

Part of the book series: Progress in Systems and Control Theory ((PSCT,volume 14))

Abstract

The analysis and design of control systems has been greatly influenced by the mathematical tools being used. Maxwell introduced linear differential equations in the 1860’s. Nyquist, Bode and others started the systematic use of tranfer functions, utilizing complex analysis in the 1930’s. Kalman brought forward state space analysis around 1960. For nonlinear systems, differential geometric concepts have been of great value recently. We will argue here that algebraic methods can be very useful for both linear and nonlinear systems. To give some motivation we will begin by looking at a few examples.

Work partially supported by the G.R. “Automatique” of the French “Centre National de la Recherche Scientifique”.

Work partially supported by the Swedish Research Council for Enigineering Sciences and by the G.R. “Automatique” of the French “Centre National de la Recherche Scientifique”.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J. Ackermann, Sampled-Data Control Systems, Springer-Verlag, Berlin, 1985.

    Book  MATH  Google Scholar 

  2. P. M. Cohn, Free Rings and their Relations, Academic Press, London, 1985.

    MATH  Google Scholar 

  3. J.M. Coron, Linearized control systems and application to smooth stabilization, SIAM J. Control Optimiz.,1993.

    Google Scholar 

  4. E. Delaleau, Lowering orders of input derivatives in generalized state representations. Proc. IFAC-Symposium NOLCOS’92, Bordeaux (1992) 209–213.

    Google Scholar 

  5. E. Delaleau and W. Respondek, Removing input derivatives and lowering their orders in generalized state-space representations. In Proc. 31st IEEE Control Decision Conf., Tucson. (1991)

    Google Scholar 

  6. E. Delaleau and M. Fliess, Algorithme de structure, filtrations et découplage, C.R. Acad. Sci. Paris, I-315, pp. 101–106, 1992.

    MathSciNet  Google Scholar 

  7. S. Diop, Elimination in Control Theory, Math. Control Signals Systems, 4, 17–32, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Diop, Differential-algebraic decision methods and some applications to system theory, Theoretical Computer Science, 98, 137–161, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  9. S. Diop and M. Fliess, On nonlinear observability, in C. Commault et al eds., Proc. 1st European Control Conf., Hermës, Paris, 152–157, 1991.

    Google Scholar 

  10. S. Diop and M. Fliess, Nonlinear observability, identifiability and persistent trajectories, in Proc. 30th IEEE Conf. on Decision and Control, IEEE Press, New York, 714–719, 1991.

    Google Scholar 

  11. M. Fliess, Automatique et corps différentiels, Forum Math., 227–238, 1989.

    Google Scholar 

  12. M. Fliess, Generalized Controller Canonical Forms for Linear and Nonlinear Dynamics, IEEE Trans. Automatic Control, AC-35, 994–1001, 1990.

    Article  MathSciNet  Google Scholar 

  13. M. Fliess, Some structural properties of generalized linear systems, Systems Control Lett, 15, 1990, pp. 391–396.

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Fliess, A simple definition of hidden modes, poles and zeros, Kybernetika, 27, pp. 186–189, 1991.

    MathSciNet  MATH  Google Scholar 

  15. M. Fliess, A remark on Willems’ trajectory characterization of linear controllability, Systems Control Lett., 19, pp. 43–45, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  16. M. Fliess and M. Hasler, Questioning the classical state space description via circuit examples. In M. A. Kashoek, J. H. van Schuppen and A. C. M. Ran, editors, Realization and Modelling in System Theory, MTNS’ 89, Birkhuser, 1990 volume 1, 1–12.

    Chapter  Google Scholar 

  17. M. Fliess, J. Lévine and P. Rouchon, A simplified approach of crane control via a generalized state-space model. Proceedings 30th IEEE Control Decision Conf., Brighton, 1991, 736–741.

    Google Scholar 

  18. M. Fliess, J. Lévine and P. Rouchon, Sur les systèmes non linéaires différentiellement plats, C. R. Acad. Sci. Paris, 315-I, 619–624, 1992.

    Google Scholar 

  19. M. Fliess, J. Lévine and P. Rouchon, Index of an implicit time-varying linear differential equation: A noncommutative linear algebraic approach. Linear Algebra Applications, to appear.

    Google Scholar 

  20. M. Fliess and F Messager, Vers une stabilisation non linéaire discontinue, in A. Bensoussan and J. L. Lions eds. Analysis and Optimization of Systems, Lect. Notes Control Inform. Sci., 144, Springer, Berlin, 778–787, 1990.

    Google Scholar 

  21. M. J. Freedmann and J. C. Willems, Smooth representation of systems with differentiated inputs, IEEE Trans. Automat. Control 23 (1978) 16–21.

    Article  Google Scholar 

  22. K. Forsman, Constructive Commutative Algebra in Nonlinear Control Theory, Linkping Studies in Science and Technology, 261, 1991.

    Google Scholar 

  23. S. T. Glad, Nonlinear state-space and input-output descriptions using differential polynomials. In J. Descusse, M. Fliess, A. Isidori and D. Leborgne, eds., New Trends in Nonlinear Control Theory, vol. 112 Lecture Notes in Control and Information Sciences, Springer verlag, (1989), 182–189.

    Chapter  Google Scholar 

  24. S. T. Glad, Differential algebraic modelling of nonlinear systems, In M. A. Kashoek, J. H. van Schuppen and A. C. M. Ran, editors, Realization and Modelling in System Theory, MTNS’ 89, Birkhuser, 1990 volume 1, 97–105.

    Chapter  Google Scholar 

  25. S. T. Glad, Nonlinear regulators and Ritt’s remainder algorithm, in Colloque International sur l’Analyse des Systèmes Dynamiques Conirolés, Lyon, July 3–6, 1990

    Google Scholar 

  26. S. T. Glad, Implementing Ritt’s algorithm of differential algebra, IFAC Symposium on Control Systems Design,NOLCOS’92, Bordeaux, France, 610–614, 1992.

    Google Scholar 

  27. S. T. Glad and L. Ljung, Model Structure Identifiability and Persistence of Excitation, Proc. 29th CDC, Honululu, Hawaii, 1990, 3236–3240.

    Google Scholar 

  28. H. M. Edwards, Galois Theory, Springer-Verlag, Berlin, 1984

    MATH  Google Scholar 

  29. M. Hasler and J. Neirynck, Circuits non linéaires,Presses Polytechniques Romandes, Lausanne, 1985; in English: Nonlinear Circuits, Artech House, Boston, 1986.

    Google Scholar 

  30. R. Hermann and A. J. Krener, Nonlinear Controllability and Observability, IEEE Trans. Automat. Control, 22 (1977) 728–740.

    Article  MathSciNet  MATH  Google Scholar 

  31. D. Hilbert, Ueber den Begriff der Klasse von Differentialgleichungen,Math. Ann., 73, 1912, pp. 95–108.

    Article  MathSciNet  MATH  Google Scholar 

  32. A. Isidori, Nonlinear Control Systems, 2nd ed., Springer-Verlag, New York, 1989.

    MATH  Google Scholar 

  33. J. Johnson, Kähler differentials and differential algebra, Ann. of Math, 89, 92–98, 1969.

    Article  MathSciNet  MATH  Google Scholar 

  34. T. Kailath, Linear systems, Prentice-Hall, Englewood Cliffs, N. J., 1980.

    MATH  Google Scholar 

  35. R. E. Kalman. Mathematical description of linear systems, SIAM J. Control, vol. 1, 152–192, 1963.

    MathSciNet  MATH  Google Scholar 

  36. R. E. Kalman. Lectures on Controllability and Observability, CIME Summer Course (Cremonese, Rome, 1968).

    Google Scholar 

  37. R. E. Kalman, P. L. Falb and M. A. Arbib, Topics in Mathematical System Theory, McGraw-Hill, New York, 1969.

    MATH  Google Scholar 

  38. E. R. Kolchin, Extensions of differential fields, III, Bull. Amer. Math. Soc., 53, 1947, 397–401.

    Article  MathSciNet  MATH  Google Scholar 

  39. E. R. Kolchin, Differential Algebra and Algebraic Groups, Academic Press, New York, 1973.

    MATH  Google Scholar 

  40. E. Kunz, Kähler differentials, Vieweg, Braunschweig/Wiesbaden, 1986.

    MATH  Google Scholar 

  41. J. W. Nieuwenhuis and J. C. Willems, Deterministic ARMA models, in A. Bensoussan and J. L. Lions eds. Analysis and Optimization of Systems, Lecture Notes Control Inform. Sci., 83, 429–439,Springerverlag, Berlin, 1986.

    Chapter  Google Scholar 

  42. H. Nijmeijer and A. van der Schaft, Nonlinear Dynamical Control Systems, Springer-Verlag, New York, 1990.

    MATH  Google Scholar 

  43. F. Ollivier, Le problème de l’identifiabilité structurelle globale: approche théorique, méthodes effectives et bornes de complexité, Thèse de Doctorat en Science, École Polytechnique, 1990.

    Google Scholar 

  44. F. Ollivier, Generalized standard bases with applications to control, Proc. European Control Conference, ECC’91, 170–176, 1991.

    Google Scholar 

  45. J. F. Pommaret, Lie Pseudogroups and Mechanics, Gordon and Breach, New York, 1988.

    MATH  Google Scholar 

  46. J. F. Ritt, Differential algebra, American Mathematical Society, Providence, RI, 1950.

    MATH  Google Scholar 

  47. C. E. Schrader and M. K. Sain, Research on system zeros: A Survey, Internat. J. Control, 50, 1407–1433, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  48. I.R. Shafarevitch, Basic Notions of Algebra, in A.I. Kostrikin and I.R. Shafarevitch eds, Algebra I, Encycl. Math. Sci., Springer-Verlag, Berlin, 1990.

    Google Scholar 

  49. E.D. Sontag, Finite dimensional open loop control generator for nonlinear control systems, Internat. J. Control, 47, pp. 537–556. 1988.

    Article  MathSciNet  MATH  Google Scholar 

  50. E.D. Sontag, Universal nonsingular controls, Systems Control Lett., 19, pp. 221–224. 1992.

    Article  MathSciNet  MATH  Google Scholar 

  51. H. J. Sussmann and V. Jurdjevic, Controllability of Nonlinear Systems, J. Dif. Equations, 12, 95–116, 1972.

    Article  MathSciNet  MATH  Google Scholar 

  52. B. L. van der Waerden, Algebra, Springer-Verlag, Berlin, 1966.

    MATH  Google Scholar 

  53. H. Weyl, The classical groups, Princeton University Press, Princeton, New Jersey, 1939.

    Google Scholar 

  54. J. C. Willems, From time series to linear systems — Part I: Finite-dimensional linear time invariant systems. Automatica 22 (1986) 561–580.

    Article  MathSciNet  MATH  Google Scholar 

  55. J. C. Willems, From time series to linear systems — Part II: Exact modelling. Automatica 22 (1986) 675–694.

    Article  MathSciNet  MATH  Google Scholar 

  56. J. C. Willems, From time series to linear systems — Part III: Approximate modelling. Automatica 23 (1987) 87–115.

    Article  MathSciNet  MATH  Google Scholar 

  57. J. C. Willems, Models for dynamics, Dynamics Reported 2, (1989) 171–269.

    MathSciNet  Google Scholar 

  58. J. C. Willems, Paradigms and puzzles in the theory of dynamical systems. IEEE Trans. Automat. Control 36 (1991) 259–294

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire des Signaux et Systemes, C.N.R.S.- E.S.E., Plateau du Moulon, 91190, Gif-sur-Yvette, France

    M. Fliess

  2. Department of Electrical Engineering, Linköping University, S-58183, Linköping, Sweden

    S. T. Glad

Authors
  1. M. Fliess
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. T. Glad
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Mathematics Institute, University of Groningen, The Netherlands

    H. L. Trentelman  & J. C. Willems  & 

Rights and permissions

Reprints and permissions

Copyright information

© 1993 Springer Science+Business Media New York

About this chapter

Cite this chapter

Fliess, M., Glad, S.T. (1993). An Algebraic Approach to Linear and Nonlinear Control. In: Trentelman, H.L., Willems, J.C. (eds) Essays on Control. Progress in Systems and Control Theory, vol 14. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0313-1_8

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-0313-1_8

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6702-7

  • Online ISBN: 978-1-4612-0313-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation