Singular Perturbations in Stable Feedback Control of Distributed Parameter Systems

  • Mark J. Balas
Chapter
Part of the International Centre for Mechanical Sciences book series (CISM, volume 280)

Abstract

In this lecture we use a singular perturbation formulation of linear time-invariant distributed parameter systems to develop a method to design finite-dimensional feedback compensators of any fixed order which will stabilize the infinite-dimensional distributed parameter system. The synthesis conditions are given entirely in terms of a finite-dimensional reduced-order model; the stability results depend on an infinite-dimensional version of the Klimushchev-Krasovskii lemma also presented here. This lecture summarizes our work on singular perturbations for stable distributed parameter system control in [9]–[10] and[24].

Keywords

Singular Perturbation Model Reduction Distribute Parameter System Uniform Asymptotic Stability Singular Perturbation Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Lions, J.L., Some Aspects of the Optimal Control of DPS, SIAM, Philadelphia, PA, 1972.Google Scholar
  2. 2.
    Ray, W.H., and Lainiotis, D., eds., DPS: Identification, Estimation, and Control, M. Dekker, NY, 1978.Google Scholar
  3. 3.
    Russell, D., “Controllability & Stabilizability Theory for Linear PDE: Recent Progress & Open Questions”, SIAM Review, 20, 371–388, 1978.CrossRefGoogle Scholar
  4. 4.
    Curtain, R., and Pritchard, A., Infinite Dimensional Systems Theory, Springer, NY, 1978.CrossRefMATHGoogle Scholar
  5. 5.
    Aziz, A., et al eds., Control Theory of Systems Governed by PDE, Academic Press, NY, 1977.Google Scholar
  6. 6.
    Balas, M., “Trends in Large Space Structure Control Theory: Fondest Hopes, Wildest Dreams”, IEEE Trans. Autom. Control Vol. 27, 522–535, 1982.CrossRefMATHGoogle Scholar
  7. 7.
    Balas, M., “Toward A (More) Practical Theory of Distributed Parameter Systèm Control”, Advances in Dynamics & Control: Theory and Applications Vol. 18, C.T. Leondes, editor, Academic Press, NY, 1982.Google Scholar
  8. 8.
    Kokotovic, P., O’Malley, R., and Sannuti, P., “Singular Perturbations & Order Reduction in Control Theory: An Overview”, Automatica, 12, 123–132, 1976.CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Balas, M., “Singular Perturbations & Model Reduction Methods for Large-Scale and Distributed Parameter Systems,” Proc. of 17th Annual Allerton Conference on Communication, Control, and Computing, Monticello, IL, 434–448, 1979.Google Scholar
  10. 10.
    Balas, M., “Reduced-Order Feedback Control of DPS Via Singular Perturbation Methods,” J. Math. Analysis & Appl. Vol. 87, 281–294, 1982.ADSCrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Balas, M., “Closed-Loop Stability of Large Space Structures Via Singular & Regular Perturbation Techniques: New Results”, Proc. of IEEE Control & Decision Conf., Albuquerque, NM, 1980.Google Scholar
  12. 12.
    Kato, T., Perturbation Theory For Linear Operators, Springer, NY, 1966.CrossRefMATHGoogle Scholar
  13. 13.
    Kwakernaak, H. & Sivan, R., Linear Optimal Control Systems, J.Wiley & Sons, NY, 1972.MATHGoogle Scholar
  14. 14.
    Curtain, R. & Pritchard, A., Functional Analysis & Modern Applied Mathematics, Academic Press, NY, 1977.MATHGoogle Scholar
  15. 15.
    Walker, J., Dynamical Systems & Evolution Equations: Theory & Applications, Plenum Press, NY, 1980.CrossRefMATHGoogle Scholar
  16. 16.
    Kimura, H., “Pole Assignment by Gain Output Feedback”, IEEE Trans. Autom. Control, 20, 509–516, 1975.CrossRefMATHGoogle Scholar
  17. 17.
    Davison, E., and Wang, S., “On Pole Assignment in Linear Multi-variable Systems Using Output Feedback”, Ibid, 516–518, 1975.Google Scholar
  18. 18.
    Vidyasagar, M., Nonlinear Systems Analysis, Prentice-Hall, NJ, 1978.Google Scholar
  19. 19.
    Klimushchev, A., and Krasovskii, N., “Uniform Asymptotic Stability of Systems of Differential Equations With A Small Parameter in the Derivative Terms”, J. Appl. Math & Mech. Vol. 25, 1961.Google Scholar
  20. 20.
    Wilde, R., and Kokotovic, P., “Stability of Singularly Perturbed Systems & Networks With Parasitics”, IEEE Trans. Autom. Contr. Vol. 17, 616–625, 1973.CrossRefMathSciNetGoogle Scholar
  21. 21.
    Zien, L., “An Upper Bound For The Singular Parameter in a Stable Singularly Perturbed System”, J. Franklin Inst. Vol. 295, 373–381, 1973.CrossRefMATHGoogle Scholar
  22. 22.
    Javid, S.H., “Uniform Asymptotic Stability of Linear Time-Varying Singularly Perturbed Systems”, Ibid, Vol. 305, 27–37, 1978.MATHGoogle Scholar
  23. 23.
    Kokotovic, P., “Subsystems, Time-Scales, & Multi-Modeling”, Proc. of IFAC Symp. on Large Scale Systems, Toulouse, France, 1980.Google Scholar
  24. 24.
    Balas, M., “Stability of Distributed Parameter Systems With Finite-Dimensional Controller-Compensators Using Singular Perturbations”, J. Math Analysis & Appl. (to appear).Google Scholar

Copyright information

© Springer-Verlag Wien 1983

Authors and Affiliations

  • Mark J. Balas
    • 1
  1. 1.Electrical, Computer, and Systems Engineering DepartmentRensselaer Polytechnic InstituteTroyUSA

Personalised recommendations