International Journal of Control, Automation, and Systems

Institute of Control, Robotics and Systems (제어로봇시스템학회)
권호 표지

Mismatching Problem between Generic Pole-assignabilities by Static Output Feedback and Dynamic Output Feedback in Linear Systems

  • Kim Su-Wood (Department of Electrical and Electronic Engineering, Cheju National University)
  • Published : 2005.03.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, it is clearly shown that the two well-known necessary and sufficient conditions mp n as generic static output feedback pole-assignment and mp + d(m+p) n+d as generic minimum d-th order dynamic output feedback pole-assignment on complex field, unbelievably, do not match up each other in strictly proper linear systems. For the analysis, a diagram analysis is newly created (which is defined by the analysis of 'convoluted rectangular/dot diagrams' constructed via node-branch conversion of the signal flow graphs of output feedback gain loops). Under this diagram analysis, it is proved that the minimum d-th order dynamic output feedback compensator for pole-assignment in m-input, p-output, n-th order systems is quantitatively decomposed into static output feedback compensator and its associated d number of arbitrary 1st order dynamic elements in augmented (m+d)-input, (p+d)-output, (n+d)-th order systems. Total configuration of the mismatched data is presented in a Table.

Keywords

References

  1. R. Hermann and C. F. Martin, 'Application of algebraic geometry to system theory. Part-I,' IEEE Trans. Automat. Control, vol. 22, pp. 19-25, 1977 https://doi.org/10.1109/TAC.1977.1101395
  2. R. W. Brockett and C. I. Byrnes, 'Multivariable Nyquist criteria, root loci and pole placement: A geometric viewpoint,' IEEE Trans. Automat. Control, vol. 26, pp. 271-284, 1981 https://doi.org/10.1109/TAC.1981.1102571
  3. X. Wang, 'Pole placement by static output feedback,' J. Math. Systems, Estimation, Control, vol. 2, pp. 205-218, 1992
  4. J. Rosenthal, 'On dynamic feedback compensation and compactification of systems,' SIAM J. Control Optim, vol. 32, pp. 279-296, 1994 https://doi.org/10.1137/S036301299122133X
  5. B. Huber and J. Verschelde, 'Pieri homotopies for problems in enumerative geometry applied to pole placement in linear systems control,' SIAM J. Control and Optimization, vol. 38, pp. 1265-1287, 2000 https://doi.org/10.1137/S036301299935657X
  6. H. Kimura, 'Pole assignment by output feedback: A longstanding open problem,' in Conf. Decision Control, vol. 12, pp. 2101-2105, 1994
  7. Y. Yang and A. L. Tits, 'Generic pole assignment may produce very fragile designs,' Proc. of 37rd conf. on Decision and Control Tampa, FL, USA, pp. 1745-1746, 1998
  8. L. Carotenuto, G. Franze and P. Muraca, 'Some results on the genericity of the pole assignment problem,' System and Control Letters, vol. 42, pp. 291-298, 2001 https://doi.org/10.1016/S0167-6911(00)00106-7
  9. X. Wang, 'Grassmannian, central projection and output feedback pole-assignment of linear systems,' IEEE Trans. Automat. Control, vol. 41, pp. 786-794, 1996 https://doi.org/10.1109/9.506231
  10. J. Rosenthal and X. Wang, 'Output feedback pole placement with dynamic compensators,' IEEE Trans. Automat. Control, vol. 41, pp. 830- 843, 1996 https://doi.org/10.1109/9.506235
  11. N. Karcanias and C. Giannakopoulos, 'Grassmann invariants, almost zeros and the determinantal zeros, pole assignment problems of linear multivariable systems,' Int. J. Control, vol. 40, pp. 673-698, 1984 https://doi.org/10.1080/00207178408933300
  12. C. Giannakopoulos and N. Karcanias, 'Pole assignment of strictly and proper linear system by constant output feedback,' Int. J. Control, vol. 42, pp. 543-565, 1985 https://doi.org/10.1080/00207178508933382
  13. R. Fuhrmann and U. Helmke, 'Output feedback invariants and canonical forms for linear dynamic systems,' Linear Circuits, Systems and Signal Processing: Theory and Application, Elsevier Science Publishers B.V. (North-Holland), pp. 279- 292, 1988
  14. M. S. Ravi, J. Rosenthal and U. Helmke, 'Output feedback invariants,' SIAM J. Control and Optimization, vol. 40, pp. 743-755, 2002
  15. G. Gratzer, General Lattice Theory, Academic Press, Inc. (London) Ltd, 1978
  16. A. G. J. McFarlane and N. Karcanias, 'Pole and zeros of linear multivariable systems: the algebraic, geometric and complex-variable theory,' Int. J. Control, vol. 24, pp. 33-74, 1976 https://doi.org/10.1080/00207177608932805