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Open problems in the area of pole placement

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Open Problems in Mathematical Systems and Control Theory

Part of the book series: Communications and Control Engineering ((CCE))

Abstract

The static and the dynamic output pole placement problem belong to the prominent design problems of modern control theory and we refer to the survey articles [4, 10, 19, 21] where also more references to the literature are provided. Various facets of the pole placement problem attracted many researchers over the years. It was been recognized right at the beginning of the problem that the output pole placement problem is nonlinear in nature and a simple solution based on techniques from linear algebra cannot be expected. In recent years significant progress has been achieved. This progress is due in a major part to a better understanding of the system theoretic ingredients and its relation to algebraic geometry. Helpful in this regard is the behavioral approach which comes in its formulation closest to the algebraic geometric nature of the pole placement problem.

Supported in part by NSF grant DMS-96-10389.

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References

  1. I. Berstein. On the Lusternik-Šnirel’mann category of real Grassmannians. Proc. Camb. Phil. Soc., 79: 129–239, 1976.

    Article  MathSciNet  MATH  Google Scholar 

  2. V. Blondel. Simultaneous Stabilization of Linear Systems, volume 191 of Lecture Notes in Control and Information Sciences. Springer-Verlag London Ltd., London, 1994.

    Google Scholar 

  3. R. W. Brockett and C. I. Byrnes. Multivariable Nyquist criteria, root loci and pole placement: A geometric viewpoint. IEEE Trans. Automat. Control, AC-26: 271–284, 1981.

    Google Scholar 

  4. C. I. Byrnes. Pole assignment by output feedback. In H. Nijmeijer and J. M. Schumacher, editors, Three Decades of Mathematical System Theory, Lecture Notes in Control and Information Sciences, volume 135, pages 31–78. Springer Verlag, 1989.

    Google Scholar 

  5. C. I. Byrnes and B. D. O. Anderson. Output feedback and generic stabilizability. SIAM J. Control Optim., 22 (3): 362–380, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  6. B. K. Ghosh. An approach to simultaneous system design, part II: Nonswitching gain & dynamic feedback compensation by algebraic geometric methods. SIAM J. Control Optim., 26 (4): 919–963, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  7. W. Helton, J. Rosenthal, and X. Wang. Matrix extensions and eigenvalue completions, the generic case. Trans. Amer. Math. Soc., 349 (8): 3401–3408, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  8. N. Karcanias and C. Giannakopoulos. Grassmann invariants, almost zeros and the determinantal zero, pole assignment problems of linear multivariable systems. Internat. J. Control, 40 (4): 673–698, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Leventides and N. Karcanias. Global asymptotic linearization of the pole placement map: A closed form solution for the constant output feedback problem. Automatica, 31 (9): 1303–1309, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  10. N. Munro. Pole assignment: A review of methods. In M. G. Singh, editor, Systems and Control Encyclopedia, pages 3710–3717. Pergamon Press, 1990.

    Google Scholar 

  11. M. S., J. Rosenthal, and X. Wang. On generic stabilizability and pole assignability. Systems & Control Letters, 23 (2): 79–84, 1994.

    Article  MathSciNet  Google Scholar 

  12. M. S. Ravi, J. Rosenthal, and X. Wang. On decentralized dynamic pole placement and feedback stabilization. IEEE Trans. Automat Contr., 40 (9): 1603–1614, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  13. M. S. Ravi, J. Rosenthal, and X. Wang. Dynamic pole assignment and Schubert calculus. SIAM J. Control Optim., 34 (3): 813–832, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  14. J. Rosenthal. On dynamic feedback compensation and compactification of systems. SIAM J. Control Optim., 32 (1): 279–296, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  15. J. Rosenthal, J. M. Schumacher, X. Wang, and J. C. Willems. Generic eigenvalue assignment for generalized linear first order systems using memoryless real output feedback. In Proc. of the 34th IEEE Conference on Decision and Control, pages 492–497, New Orleans, Louisiana, 1995.

    Google Scholar 

  16. J. Rosenthal, J. M. Schumacher, and J. C. Willems. Generic eigenvalue assignment by memoryless real output feedback. Systems & Control Letters, 26: 253–260, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  17. J. Rosenthal and F. Sottile. Some remarks on real and complex output feedback. Systems & Control Letters, 33 (2): 73–80, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  18. J. Rosenthal and X. Wang. Output feedback pole placement with dynamic compensators. IEEE Trans. Automat. Contr., 41 (6): 830–843, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  19. J. Rosenthal and X. Wang. Inverse eigenvalue problems for multi- variable linear systems. In C. I. Byrnes, B. N. Datta, D. Gilliam, and C. F. Martin, editors, Systems and Control in the Twenty-First Century, pages 289–311. Birkäuser, 1997.

    Google Scholar 

  20. H. Schubert. Beziehungen zwischen den linearen Räumen auferleg¬baren charakteristischen Bedingungen. Math. Ann., 38: 598–602, 1891.

    Article  MathSciNet  Google Scholar 

  21. V. L. Syrmos, C. T. Abdallah, P. Dorato, and K. Grigoriadis. Static output feedback—a survey. Automatica, 33 (2): 125–137, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  22. X. Wang. Pole placement by static output feedback. Math. Systems, Estimation, and Control, 2 (2): 205–218, 1992.

    Google Scholar 

  23. X. Wang. Grassmannian, central projection and output feedback pole assignment of linear systems. IEEE Trans. Automat. Contr., 41 (6): 786–794, 1996.

    Article  MATH  Google Scholar 

  24. J. C. Willems. Paradigms and puzzles in the theory of dynamical systems. IEEE Trans. Automat. Control, AC-36(3): 259–294, 1991.

    Google Scholar 

  25. J. C. Willems. On interconnections, control, and feedback. IEEE Trans. Automat Control, 42 (3): 326–339, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  26. J. C. Willems. Generic eigenvalue assignability by real memoryless output feedback made simple. A. Paulraj, V. Roychowdhury, and C.D. Schaper, editors, In Communications, Computation, Control and Signal Processing, Kluwer, pages 343–354, 1997.

    Google Scholar 

  27. J. C. Willems and W. H. Hesselink. Generic properties of the pole placement problem. In Proc. of the 7th IFAC Congress, pages 1725–1729, 1978.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Notre Dame, Notre Dame, IN, 46556, USA

    Joachim Rosenthal

  2. Mathematics Institute, University of Groningen, P.O. Box 800, 9700 AV, Groningen, The Netherlands

    Jan C. Willems

Authors
  1. Joachim Rosenthal
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  2. Jan C. Willems
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Editor information

Editors and Affiliations

  1. Institute of Mathematics, University of Leige, Sart Tilman B37, B-4000, Liege, Belgium

    Vincent Blondel (PhD) (PhD)

  2. Department of Mathematics, Rutgers University, Hill Center 724, 08903, New Brunswick, USA

    Eduardo D. Sontag (PhD) (PhD)

  3. Centre for Arificial Intelligence and Robotics, Raj Bhavan Cirlce - High Grounds, 560001, Bangalore, India

    Mathukumalli Vidyasagar (PhD) (PhD)

  4. Institute of Mathematics, Rijksuniversiteit te Groningen, Postbus 800, 9700, Av Groningen, The Netherlands

    Jan C. Willems (PhD) (PhD)

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© 1999 Springer-Verlag London Limited

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Rosenthal, J., Willems, J.C. (1999). Open problems in the area of pole placement. In: Blondel, V., Sontag, E.D., Vidyasagar, M., Willems, J.C. (eds) Open Problems in Mathematical Systems and Control Theory. Communications and Control Engineering. Springer, London. https://doi.org/10.1007/978-1-4471-0807-8_37

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  • DOI: https://doi.org/10.1007/978-1-4471-0807-8_37

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-4471-1207-5

  • Online ISBN: 978-1-4471-0807-8

  • eBook Packages: Springer Book Archive

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