Skip to main content
SpringerLink
Log in

State feedback and estimation of well-posed systems

  • Published:
Mathematics of Control, Signals and Systems Aims and scope Submit manuscript

Abstract

The class of well-posed systems includes many systems modeled by partial differential equations with boundary control and point sensing as well as many other systems with possibly unbounded control and observation. The closed-loop system created by applying state-feedback to any well-posed system is well-posed. A state-space realization of the closed loop is derived. A similar result holds for state estimation of a well-posed system. Also, the classical state-feedback/estimator structure extends to well-posed systems. In the final section state-space realizations for a doubly coprime factorization for well-posed systems are derived.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. H. T. Banks and K. A. Morris, Input-output stability for accelerometer control systems,Control-Theory Adv. Technol.,10 (1994), 1–17.

    Google Scholar 

  2. J. Bontsema and R. F. Curtain, Perturbation properties of a class of infinite-dimensional systems with unbounded control and observation,IMA J. Math. Control Inform.,5 (1988), 333–352.

    Google Scholar 

  3. R. F. Curtain, Equivalence of input-output stability and exponential stability for infinitedimensional systems,Math. Systems Theory,21 (1988), 19–48.

    Google Scholar 

  4. R. F. Curtain, Equivalence of input-output stability and exponential stability,Systems Control Lett.,12 (1989), 235–239.

    Google Scholar 

  5. R. F. Curtain, H. Logemann, S. Townley, and H. Zwart, Well-posedness, stabilizability and admissibility for Pritchard-Salamon systems,Math. Systems Estimation Control, to appear.

  6. R. F. Curtain and A. J. Pritchard,Infinite-Dimensional Linear Systems Theory, Springer-Verlag, Berlin, 1978.

    Google Scholar 

  7. R. F. Curtain and A. J. Pritchard, Robust Stabilization of Infinite-Dimensional Systems with Respect to Coprime Factor Perturbations, Technical Report W-9220, Department of Mathematics, University of Groningen, 1992.

  8. R. F. Curtain and G. Weiss, Well-posedness of triples of operators (in the sense of linear systems theory), inControl and Estimation of Distributed Parameter Systems, Vorau, July 10–16, 1988 (F. Kappel, K. Kunisch, and W. Schappacher, eds.), pp. 41–59, International Series of Numerical Mathematics, vol. 91, Birkhauser Verlag, Basel, 1989.

    Google Scholar 

  9. R. Datko, Extending a theorem of A. M. Liapunov to Hilbert space,J. Math. Anal. Appl,32 (1970), 610–616.

    Google Scholar 

  10. J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis, State-space solutions to standard H2 and H∞ control problems,IEEE Trans. Automat. Control,34 (1989), 831–847.

    Google Scholar 

  11. C. A. Jacobson and C. N. Nett, Linear state-space systems in infinite-dimensional space: The role and characterization of joint stabilizability/detectability,IEEE Trans. Automat. Control,33 (1988), 541–549.

    Google Scholar 

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

    Google Scholar 

  13. B. van Keulen, A state-space approach to H∞-control problems for infinite-dimensional systems, inAnalysis and Optimization of Systems: State and Frequency Domain Approaches for Infinite-Dimensional Systems, Sophia-Autipolis, June 9–12, 1992 (R. F. Curtain, ed.), pp. 46–71, Lecture Notes in Control and Information Sciences, vol. 185, Springer-Verlag, Berlin, 1992.

    Google Scholar 

  14. P. P. Khargonekar and E. D. Sontag, On the relation between stable matrix fraction factorizations and regulable realizations of linear systems over rings,IEEE Trans. Automat. Control,27 (1982), 627–638.

    Google Scholar 

  15. H. Logemann, On the transfer matrix of a neutral system: Characterizations of exponential stability in input-output terms,Systems Control Lett.,9 (1987), 393–400.

    Google Scholar 

  16. K. A. Morris, The well-posedness of accelerometer control systems, inAnalysis and Optimization of Systems: State and Frequency Domain Approaches for Infinite-Dimensional Systems, Sophia-Antipolis, June 9–12, 1992 (R. F. Curtain, ed.), pp. 378–387, Lecture Notes in Control and Informatio

  17. K. A. Morris, State-Space Realizations of Coprime Factorizations of Well-posed Systems, Technical Report FI93-CT08, Fields Institute, Waterloo, 1993.

    Google Scholar 

  18. K. A. Morris, Perturbation of well-posed systems by state-feedback, inIdentification and Control in Systems Governed by Partial Differential Equations, Mount Holyoke, July 11–16, 1992 (H. T. Banks, R. H. Fabiano, and K. Ito, eds.), pp. 141–154, SIAM, Philadelphia, 1993.

    Google Scholar 

  19. C. N. Nett, C. A. Jacobson, and M. J. Balas, A connection between state-space and doubly coprime fractional representations,IEEE Trans. Automat. Control,29 (1984), 831–832.

    Google Scholar 

  20. A. Pazy,Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.

    Google Scholar 

  21. A. J. Pritchard and D. Salamon, The linear quadratic control problem for infinite dimensional systems with unbounded input and output operators,SIAM J. Control Optim.,25 (1987), 121–144.

    Google Scholar 

  22. D. Salamon,Control and Observation of Neutral Systems, Pitman, Boston, 1984.

    Google Scholar 

  23. D. Salamon, Infinite-dimensional linear systems with unbounded control and observation: A functional analytic approach,Trans. Amer. Math. Soc.,300 (1987), 383–431.

    Google Scholar 

  24. D. Salamon, Realization theory in Hilbert space,Math. Systems Theory,21 (1989), 147–164.

    Google Scholar 

  25. M. C. Smith, On stabilization and the existence of coprime factorizations,IEEE Trans. Automat. Control,34 (1989), 1005–1007.

    Google Scholar 

  26. M. Vidyasagar,Control System Synthesis: A Factorization Approach, MIT Press, Cambridge, MA, 1985.

    Google Scholar 

  27. G. Weiss, Admissibility of unbounded control operators,SIAM J. Control Optim.,27 (1989), 527–545.

    Google Scholar 

  28. G. Weiss, Admissible observation operators for linear semigroups,Israel J. Math.,64 (1989), 17–43.

    Google Scholar 

  29. G. Weiss, The representation of regular linear systems on Hilbert spaces, inControl and Estimation of Distributed Parameter Systems, Vorau, July 10–16, 1988 (F. Kappel, K. Kunisch, and W. Schappacher, eds.), pp. 401–416, International Series of Numerical Mathematics, vol. 91, Birkhauser Verlag, Basel, 1989.

    Google Scholar 

  30. G. Weiss, Two conjectures on the admissibility of control operators, inControl and Estimation of Distributed Parameter Systems, Vorau, July 8–14, 1990 (F. Kappel, K. Kunisch, and W. Schappacher, eds.), pp. 367–378, International Series of Numerical Mathematics, vol. 100, Birkhauser Verlag, Basel, 1991.

    Google Scholar 

  31. G. Weiss, Regular linear systems with feedback,Math. Control Signals Systems,7 (1994), 23–57.

    Google Scholar 

  32. A. H. Zemanian,Distribution Theory and Transform Analysis, Dover, New York, 1965.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, University of Waterloo, N2L 3G1, Waterloo, Ontario, Canada

    K. A. Morris

Authors
  1. K. A. Morris
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This research was partially supported by the Fields Institute, which is funded by grants from the Ontario Ministry of Colleges and Universities and the Natural Sciences and Engineering Research Council of Canada, and by a grant from the Natural Sciences and Engineering Research Council of Canada.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Morris, K.A. State feedback and estimation of well-posed systems. Math. Control Signal Systems 7, 351–388 (1994). https://doi.org/10.1007/BF01211524

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01211524

Key words

Search

Navigation