Skip to main content
SpringerLink
Log in

A Separation Principle for Nonlinear Systems

  • Chapter
  • First Online:
Book cover Algorithms of Estimation for Nonlinear Systems

Part of the book series: Understanding Complex Systems ((UCS))

  • 1161 Accesses

Abstract

Since the early 1990s, a variety of approaches have been proposed for the synthesis of observers and controllers for nonlinear systems. A considerable number of researchers have studied the stability and asymptotic output tracking problems from different perspectives. An appealing approach is based on differential-geometric methods that are summarized in Isidori’s outstanding book [24]. In isidori’s treatment, a clear connection is established with the concepts of the inverse system and the zero dynamics using the notion of relative degree or relative order and the associated normal canonical form for nonlinear systems [2, 24]. This was an interesting generalization to the problem of exactly linearizing a nonlinear control system by means of a static-state feedback, which was solved independently by Jakubczyk and Respondek [25] and by Hunt et al. [23]. We refer to the references [5, 24] for a survey on this topic and for some material on input–output linearization. Over the past three decades, Charlet et al. [4] have tried to weaken the aforementioned conditions by allowing dynamic state feedbacks. They were able to prove, among other things, that for single-input systems, dynamic and static feedback condition coincide. In [28], interesting results on output stabilization for observed nonlinear systems via dynamic output feedback were considered, allowing one to deal with singularities that can appear. On the other hand, very important contributions have been made by Fliess and coworkers [9–19] using techniques based on differential algebra. Fliess’s ideas have contributed to a revision and clarification of the deeply rooted state-space approach [18, 42]. This approach has succeeded in clearly establishing basic concepts such as controllability, observability, invertibility, model matching, realization, exact linearization , and decoupling. Within this viewpoint, canonical forms [7–19, 31–34, 38–45] for nonlinear controlled systems are allowed to explicitly exhibit time derivatives of the control input functions on the state and output equations. Elimination of these input derivatives from the state equations via control-dependent state coordinate transformations is possible in the case of linear systems.

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 89.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 119.99
Price excludes VAT (USA)
  • Durable hardcover 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

References

  1. Amira DTS200: Laboratory Setup Three Tank System. Amira Gmbh, Duisburgh, Germany (1996)

    Google Scholar 

  2. Byrnes, C.I., Isidori, A.: A frequency domain philosophy for nonlinear systems. In: Proceedings 23rd IEEE Conference Decision and Control, 1569–1573, 1984

    Google Scholar 

  3. Castillo, B., Poznyak, A., Lopez, V.: Approximate tracking near singularities. A differential algebraic approach. In: Proceedings of the 32nd IEEE Conference on Decision and Control, pp. 2778–2782, 1993

    Google Scholar 

  4. Charlet, B., Levine, J., Marino, R.: On dynamic feedback linearization. Syst. Cont. Lett. 13, 143–151 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  5. Claude, D.: Everything you always wanted to know about linearization, In: Fliess, M., Hazewinkel, M. (eds), Algebraic and Geometric Method in Nonlinear Control Theory, pp. 181–220. Reidel, Dordrecht (1986)

    Chapter  Google Scholar 

  6. Dayawansa, W.P., Martin, C.F., Knowles, G.: A global nonlinear tracking problem. In: Proceedings of the 29th IEEE Conference on Decision and Control, pp. 1268–1271, 1992

    Google Scholar 

  7. De León-Morales, J., Alvarez-Leal, J.G., Martínez-Guerra, R.: A dynamical linearizing feedback controller for robots with flexible joint. In: Proceedings IEEE Conference on Control Applications, pp. 306–311, 1997

    Google Scholar 

  8. Deleleau, E., Respondek, W.: Lowering the orders of derivatives of controls in generalized state-space systems. J. Math. Syst. Estim. Control 8, 427–453 (1998)

    MathSciNet  Google Scholar 

  9. Diop, S.. Fliess, M.: On nonlinear observability. In: Commault, C., et al. (eds.), Proceedings of the First European Control Conference, pp. 152–157. Hermes, Grenoble, Paris (1991)

    Google Scholar 

  10. Fliess, M.: Generalized controller canonical forms for linear and nonlinear dynamics. IEEE Trans. Automatic Control 35 (9), 994–1001 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fliess, M.: Quelques remarques sur les observateurs nonlinéaires. In: Proc. Colloque Gretsi Trait. Signal Images, Nice, pp. 169–172, 1987

    Google Scholar 

  12. Fliess, M.: Nonlinear control theory and differential algebra. In: Byrnes, Ch.I., Kurzhanski, A. (eds.), Modelling and Adaptive Control (Lecture Notes Control Inform. Sci.), vol. 105, pp. 134–145. Springer, New York (1988)

    Chapter  Google Scholar 

  13. Fliess, M.: Géneralisation non linéaire de la forme canonique de commmande et linéarisation par bouclage. C.R. Acad. Sci. Paris I-308, 377–379 (1989)

    MathSciNet  MATH  Google Scholar 

  14. Fliess, M.: Automatique et corps différentiels. Forum Math. 1, 227–238 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fliess, M.A.: A note on the invertibility of nonlinear input–output systems. Syst. Control Lett. 8, 147–151 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fliess, M.: What the Kalman state variable representation is good for? In: 29th IEEE Conference on Decision and Control Honolulu, Hawaii, vol. 3, pp. 1282–1287, 1990

    Google Scholar 

  17. Fliess, M., Chantre, P., Abu el Ata, S., Coic, A.: Discontinuous predictive control, inversion and singularities; application to a heat exchanger. In: Proceedings of the International Conference on Analysis and Optimization of Systems, Antibes, France. Lecture Notes in Control and Information Sciences. Springer, Berlin (1990)

    Google Scholar 

  18. Fliess, M, Hassler, M.: Questioning the classic state-space description via circuit examples. In: Kaashoek, M.A., Ram, A.C.M., Van Schuupen. J.H. (eds), Mathematical Theory of Networks and Systems, MTNS-89, Progress in Systems and Control. Birkhäuser, Boston (1990)

    Google Scholar 

  19. Fliess, M., Messager, F.: Vers une stabilisation non linéaire discontinue. In Bensousan, A., Lions, J.L. (eds.), Analysis Optimization of Systems, Lecture Notes in Control and Information Sciences, vol. 144, pp. 778–787. Springer, Berlin (1990)

    Google Scholar 

  20. Fliess, M., Join, C., Sira-Ramirez, H.: Closed-loop fault tolerant control for uncertain nonlinear systems. In: Meurer, T., et al. (eds.): Control and Observer Design, LNCIS 322, pp. 217–233. Springer, Berlin/Heidelberg (2005)

    Google Scholar 

  21. Gauthier, J.P., Hammouri, H., Othman, S.: A simple observer for nonlinear systems applications to bioreactors. IEEE Trans. Automatic Control 36 (6), 875–880 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hunt, L.R., Meyer, G., Ramakrishna, V.: Output tracking and steady state for nonlinear systems. In: 35 th IEEE Conference on Decision and Control, Kobe, Japan, pp. 2064–2068, 1996

    Google Scholar 

  23. Hunt, L.R., Su, R., Meyer, G.: Global transformation of nonlinear systems. IEEE Trans. Automatic Control AC-28, 24–31 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  24. Isidori, A.: Nonlinear Control Systems, 3rd edn. Springer, New York (1991)

    MATH  Google Scholar 

  25. Jakubczyk, B., Respondek, W.: On linearization of control systems. Bull. Acad. Polon. Scie. Ser. Math. 28, 517–522 (1980)

    MathSciNet  MATH  Google Scholar 

  26. Join, C., Ponsart, J.C., Sauter, D., Theilliol, D.: Nonlinear filter design for fault diagnosis: application to the three-tank system. IEE Proc. Control Theory Appl. 152 (1), 55–64 (2005)

    Article  Google Scholar 

  27. Join, C., Sira-Ramírez, H., Fliess, M.: Control of an uncertain three tank system via on-line parameter identification and fault detection. In: Proc. of 16th Triennial World IFAC Conference (IFAC’05), Prague, Czech Republic, 2005

    Google Scholar 

  28. Jouan, Ph., Gauthier, J.P.: Finite singularities of nonlinear systems, output stabilization and observers. J. Dyn Control Syst. 2, 252–288 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kang, W., Krener, A.J.: Nonlinear observer design a backstepping approach, preprint, 1998

    Google Scholar 

  30. Lapidus, L., Amundson, N.R.: Chemical Reactor Theory: A Review. Prentice-Hall,, Englewood Cliffs (1977)

    Google Scholar 

  31. Martínez-Guerra, R., De León-Morales, J.: Observers for a multi-input multi-output bilinear system class: a differential algebraic approach. J. Math. Comput. Model. 20 (12), 125–132 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  32. Martínez-Guerra, R., De León-Morales, J.: Nonlinear estimators: a differential algebraic approach. J. Appl. Math.Lett. 9 (4), 21–25 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  33. Martínez-Guerra, R., De León-Morales, J., Huerta-Guevara, O.: Observer-based controller for a synchronous generator. In: 38th IEE Conference on Decision and Control, Phoenix, Arizona, pp. 4644–4649, 1999

    Google Scholar 

  34. Martínez-Chitoy, C., De León-Morales, J., Martínez-Guerra, R.: Estimation and nonlinear control of a class of continuous stirred tank reactors: a differential algebraic approach. In: Proceedings of the American Control Conference, 3390–3394, 1997

    Google Scholar 

  35. Martínez-Guerra, R., Mata-Machuca, J.L.: Fault Detection and Diagnosis in Nonlinear Systems: A Differential and Algebraic Viewpoint. Springer, New York (2014)

    Book  MATH  Google Scholar 

  36. Martínez-Guerra, R., Mata-Machuca, J.L., Rincon-Pasaye, J.J.: Fault diagnosis viewed as a left invertibility problem. ISA Trans. 52, 652–661 (2013)

    Article  Google Scholar 

  37. Nijmeijer, H., Van Der Schaft, A.J.: Nonlinear Dynamical Control Systems. Springer, New York (1990)

    Book  MATH  Google Scholar 

  38. Rudolph, J., Birk, J., Zeitz, M.: Dependence on time derivatives of the input in the nonlinear controller canonical form. In: Conte, G., Perdon, A., Wyman, W. (eds.), New Trends in Systems Theory, pp. 636–643. Boston, Birkhäuser (1991)

    Chapter  Google Scholar 

  39. Sira-Ramírez, H., Ahmad, S., Zribi, N.: Dynamical feedback control of robotic manipulator with joint flexibility. IEEE Trans. Syst. Man Cybern. 22 (4), 736–747 (1992)

    Article  MATH  Google Scholar 

  40. Sira-Ramírez, H.: The differential algebraic approach in nonlinear dynamical feedback controlled landing maneuvers. IEEE Trans. Automatic Control 37, 518–524 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  41. Sira-Ramírez, H.: Dynamical feedback strategies in aerospace systems control: a differential algebraic approach. In: Proc. Eur. Contr. Conf., Grenoble, France, vol. 3, pp. 2238–2243, 1991

    Google Scholar 

  42. Sira-Ramírez, H.: Dynamical pulse width modulation control of nonlinear systems. Syst. Control Lett. 18, 223–231 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  43. Sira-Ramírez, H.: Nonlinear dynamical discontinuous feedback controlled descent on a non atmosphere-free planet: a differential algebraic approach. Control Theory Adv. Tech. 7 (2), 301–320 (1991)

    MathSciNet  Google Scholar 

  44. Sira-Ramírez, H.: On the robust design of sliding observers for linear systems. Syst. Control Lett. 23, 9–14 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  45. Sira-Ramírez, H, Lischinsky-Arenas, P.: The differential algebraic approach in nonlinear dynamical compensator design for DC-to-DC power converters. Int. J. Control 54 (1), 111–134 (1991)

    Article  Google Scholar 

  46. Teel, A., Praly, L.: Tool for semiglobal stabilization by partial state and output feedback. SIAM J. Control Optim. 33 (5), 1443–1448 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  47. Trejo-Zúñiga, I., Martínez-Guerra, R.: An invariant observer for fault diagnosis: a real-time application, XXI Congreso de la Asociacion Chilena de Control Automático ACCA 2014, Santiago de Chile, 5–7 de Noviembre, pp. 393–398, 2014

    Google Scholar 

  48. Vidyasagar, M.: Decomposition techniques for large-scale system with non-additive interactions: stability and stabilizability. IEEE Trans. Automatic Control 25 (4), 773–779 (1980)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Control Automatico, CINVESTAV-IPN, Mexico, Distrito Federal, Mexico

    Rafael Martínez-Guerra & Christopher Diego Cruz-Ancona

Authors
  1. Rafael Martínez-Guerra
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christopher Diego Cruz-Ancona
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Martínez-Guerra, R., Cruz-Ancona, C.D. (2017). A Separation Principle for Nonlinear Systems. In: Algorithms of Estimation for Nonlinear Systems. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-53040-6_9

Download citation

Publish with us

Policies and ethics

Search

Navigation