Optimization and pole assignment in control system design

Eric Chu

International Journal of Applied Mathematics and Computer Science (2001)

  • Volume: 11, Issue: 5, page 1035-1053
  • ISSN: 1641-876X

Abstract

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Some elementary optimization techniques, together with some not so well-known robustness measures and condition numbers, will be utilized in pole assignment. In particular, ''Method 0'' by Kautsky et al. (1985) for optimal selection of vectors is shown to be convergent to a local minimum, with respect to the condition number . This contrasts with the misconception by Kautsky et al. that the method diverges, or the recent discovery by Yang and Tits (1995) that the method converges to stationary points.

How to cite

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Chu, Eric. "Optimization and pole assignment in control system design." International Journal of Applied Mathematics and Computer Science 11.5 (2001): 1035-1053. <http://eudml.org/doc/207544>.

@article{Chu2001,
abstract = {Some elementary optimization techniques, together with some not so well-known robustness measures and condition numbers, will be utilized in pole assignment. In particular, ''Method 0'' by Kautsky et al. (1985) for optimal selection of vectors is shown to be convergent to a local minimum, with respect to the condition number . This contrasts with the misconception by Kautsky et al. that the method diverges, or the recent discovery by Yang and Tits (1995) that the method converges to stationary points.},
author = {Chu, Eric},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {condition number; pole assignment; robustness; linear feedback; computational methods; pole placement; optimization; modal matrix},
language = {eng},
number = {5},
pages = {1035-1053},
title = {Optimization and pole assignment in control system design},
url = {http://eudml.org/doc/207544},
volume = {11},
year = {2001},
}

TY - JOUR
AU - Chu, Eric
TI - Optimization and pole assignment in control system design
JO - International Journal of Applied Mathematics and Computer Science
PY - 2001
VL - 11
IS - 5
SP - 1035
EP - 1053
AB - Some elementary optimization techniques, together with some not so well-known robustness measures and condition numbers, will be utilized in pole assignment. In particular, ''Method 0'' by Kautsky et al. (1985) for optimal selection of vectors is shown to be convergent to a local minimum, with respect to the condition number . This contrasts with the misconception by Kautsky et al. that the method diverges, or the recent discovery by Yang and Tits (1995) that the method converges to stationary points.
LA - eng
KW - condition number; pole assignment; robustness; linear feedback; computational methods; pole placement; optimization; modal matrix
UR - http://eudml.org/doc/207544
ER -

References

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  1. Andry A.N., Shapiro E.Y. and Chung J.C. (1983): Eigenstructure assignment for linear systems. — IEEE Trans. Aerosp. Electr. Syst., Vol.19, pp.711–729. 
  2. Bertsekas D.P. (1995): Nonlinear Programming. — Belmont: Athena Scientific. 
  3. Bhattacharyya S.P. and De Sousa E. (1982): Pole assignment via Sylvester equations. — Syst. Contr. Lett., Vol.1, pp.261–263. Zbl0473.93037
  4. Boyd S., El Ghaoui L., Feron E. and Balakrishnan V. (1994): Linear Matrix Inequalities in System and Control Theory. — Philadelphia: SIAM. Zbl0816.93004
  5. Brogan W.L. (1974): Applications of a determinant identity to pole-placement and observer problems. — IEEE Trans. Automat. Contr., Vol.19. Zbl0291.93021
  6. Byers R. and Nash S.G. (1989): Approaches to robust pole assignment. — Int. J. Contr., Vol.49, pp.97–117. Zbl0666.93042
  7. Cavin K.R. and Bhattacharyya S.P. (1982): Robust and well-conditioned eigenstructure assignment via Sylvester’s equation. — Proc. Amer. Contr. Conf.. Zbl0512.93035
  8. Chu E.K. (1986a): A pole-assignment algorithm for linear state feedback. — Syst. Contr. Lett., Vol.7, pp.289–299. Zbl0592.93024
  9. Chu E.K. (1986b): Generalizations of the Bauer-Fike theorems. — Numer. Math., Vol.49, pp.85–91. 
  10. Chu E.K. (1987): Exclusion theorems and perturbation theory for the generalized eigenvalue problem. — SIAM J. Numer. Anal., Vol.24, pp.1114–1125. Zbl0636.15009
  11. Chu E.K. (1988): A controllability condensed form and a state feedback pole assignment algorithm for descriptor systems. — IEEE Trans. Automat. Contr., Vol.33, pp.366–370. Zbl0635.93033
  12. Chu E.K. (1993): Approximate pole assignment. — Int. J. Contr., Vol.59, pp.471–484. Zbl0781.93036
  13. Chu E.K. (2001a): Optimization and pole assignment in control system design. — Reprint & Preprint Series, Dept. Math. Stat., Monash University. 
  14. Chu E.K. (2001b): Pole assignment for second-order systems. — Mech. Syst. Signal Process., (to appear). 
  15. Chu E.K. and Datta B.N. (1996): Numerically robust pole assignment for second-order systems. — Int. J. Contr., Vol.64, pp.1113–1127. Zbl0850.93318
  16. Chu E.K. and Li N. (1993): Controllability measures and their computation. — CTAC-91, pp.291–298. 
  17. Chu E.K. and Li N. (1994): Designing the Hopfield neural network via pole assignment. — Int. J. Syst. Sci., Vol.25, pp.669–681. Zbl0805.93020
  18. Cichocki A. and Unbehauen R. (1993): Neural Networks for Optimization and Signal Processing. — London: Wiley. Zbl0824.68101
  19. Datta B.N. and Saad Y. (1991): Arnoldi methods for large Sylvester-like observer matrix equations and an associate algorithm for partial pole assignment. — Lin. Alg. Applic., Vol.154–156, pp.225–244. 
  20. Davidon W.C. (1975): Optimally conditioned optimization algorithms without line searches. — Math. Prog., Vol.9, pp.1–30. Zbl0328.90055
  21. Dennis J.E., Jr., and Schnabel R.B. (1983): Numerical Methods for Unconstrained Optimization and Nonlinear Equations. — Englewood Cliffs: Prentice-Hall. 
  22. Dennis J.E. and Wolkowicz H. (1990): Sizing and least change secant methods. — Tech. Rep., Vol.COOR 90–02, Dept. Combinatorics and Optimization, University of Waterloo, Ontario, 1990. 
  23. Fahmy M.M. and O’Reilly J. (1982): On eigenstructure assignment in linear multivariable system. — IEEE Trans. Automat. Contr., Vol.27. 
  24. Fahmy M.M. and O’Reilly J. (1988): Multistage parametric eigenstructure assignment by output feedback. — Int. J. Contr., Vol.48, pp.97–116. Zbl0688.93023
  25. Fahmy M.M. and O’Reilly J. (1988): Parametric eigenstructure assignment by output feedback control. — Int. J. Contr., Vol.48, pp.1519–1535. Zbl0658.93043
  26. Fletcher R. (1987): Practical Methods of Optimization, 2-nd Ed. — Chichester: Wiley. 
  27. Golub G.H. and Van Loan C.F. (1989): Matrix Computations, 2-nd Ed. — Baltimore: Johns Hopkins University Press. 
  28. Gourishanker V. and Ramar K. (1976): Pole assignment with minimum eigenvalue sensitivity to plant variations. — Int. J. Contr., Vol.23, pp.493–504. Zbl0317.93031
  29. He C., Laub A. and Mehrmann V. (1995): Placing plenty of poles is pretty. Preposterous. — Vol.SPC 95–17, Forschergruppe “Scientific Parallel Computing”, Fakultät für Mathematik, TU Chemnitz-Zwickau, FRG. 
  30. Ho D., Lam J., Xu J. and Tam H.K. (1996): Recurrent neural networks for output feedback robust approximate pole assignment. — Res. Rep., Vol.MA–96–08, Faculty of Science and Technology, City University of Hong Kong. 
  31. Joshi S.M. (1989): Control of Large Flexible Space Structures. — Berlin: Springer. Zbl0762.93001
  32. Karmarkar N. (1984): A new polynomial-time algorithm for linear programming. — Combinatorica, Vol.4, pp.373–395. Zbl0557.90065
  33. Katti S.K. (1983): Pole assignment in multi-input systems via elementary transformations. — Int. J. Contr., Vol.37, pp.315–347. Zbl0502.93029
  34. Kautsky J., Nichols N.K. and Van Dooren P. (1985): Robust pole assignment in linear state feedback. — Int. J. Contr., Vol.41, pp.1129–1155. Zbl0567.93036
  35. Kimura H. (1975): Pole assignment by gain output feedback. — IEEE Trans. Automat. Contr., Vol.20, pp.509–516. Zbl0309.93017
  36. Kimura H. (1977): A further result on the problem of pole assignment by output feedback. — IEEE Trans. Automat. Contr., Vol.22, pp.458–463. Zbl0355.93008
  37. Klein G. and Moore B.C. (1977): Eigenvalue-generalized eigenvector assignment with state feedback. — IEEE Trans. Automat. Contr., Vol.22, pp.140–141. Zbl0346.93020
  38. Lam J. and Yan W.Y. (1995): A gradient flow approach to robust pole-assignment problem. — Int. J. Robust Nonlin. Contr., Vol.5, pp.175–185. Zbl0824.93027
  39. Marcus M. (1962): An inequality connecting the p-condition number and the determinant. — Numer. Math., Vol.4, pp.350–353. Zbl0113.32003
  40. Math Works (1995): MATLAB Version 4 Users’ Guide. — Englewood Cliffs: Prentice Hall. 
  41. Mayne D. and Murdoch P. (1970): Model control of linear time invariant systems. — Int. J. Contr., Vol.11. Zbl0186.48204
  42. Miminis G.S. (1981): Numerical Algorithms for Controllability and Eigenvalue Allocation. — Ph.D. Thesis, School Comp. Sci., McGill University. 
  43. Miminis G.S. and Paige C.C. (1982a): An algorithm for pole assignment of time invariant multi-input linear systems. — Proc. IEEE Conf. Decision Control, pp.62–67. Zbl0478.93022
  44. Miminis G.S. and Paige C.C. (1982b): An algorithm for pole assignment of time-invariant linear systems. — Int. J. Contr., Vol.35, pp.341–354. Zbl0478.93022
  45. Miminis G.S. and Paige C.C. (1988): A direct algorithm for pole assignment of time-invariant multi-input linear systems using state feedback. — Automatica, Vol.24. Zbl0652.93015
  46. Paige C.C. (1981): Properties of numerical algorithms related to computing controllability. — IEEE Trans. Automat. Contr., Vol.26, pp.130–138. Zbl0463.93024
  47. Petkov P.Hr., Christov N.D. and Konstantinov M.M. (1986): A computational algorithm for pole assignment of linear multi-input systems. — IEEE Trans. Automat. Contr., Vol.31, pp.1044–1047. Zbl0607.93020
  48. Rosenbrock H.H. (1970): State-Space and Multivariable Theory. — London: Nelson. Zbl0246.93010
  49. Saad Y. (1988): Projection and deflation methods for partial pole assignment in linear state feedback. — IEEE Trans. Automat. Contr., Vol.33, pp.290–297. Zbl0641.93031
  50. Stewart G.W. and Sun J.G. (1990): Matrix Perturbation Theory. — San Diego: Academic Press. 
  51. Varga A. (1981a): A Schur method for pole assignment. — IEEE Trans. Automat. Contr., Vol.26, pp.517–519. Zbl0475.93040
  52. Varga A. (1981b): Numerical stable algorithm for standard controllability form determination. — Electr. Lett., Vol.17, pp.74–75. 
  53. Wolkowicz H. (1990): Measures for symmetric rank-one updates. — Tech. Rep., Vol.CORR 90–03, Dept. Combinat. Optim., University of Waterloo, Ontario. 
  54. Wonham W.M. (1979): Linear Multivariable Control: A Geometric Approach, 2-nd Ed. — Berlin: Springer. Zbl0424.93001
  55. Xiao Y., Crusca F. and Chu E.K. (1996): Bilinear matrix inequalities in robust control: phase I—Problem formulation. — Tech. Rep., Vol.TR–96–3, Dept. Electr. Comp. Syst. Eng., Monash University, Caulfield 3145, Australia. 
  56. Yang Y. (1989): A new condition number of eigenvalue and its application in control theory. — J. Computat. Math., Vol.7, pp.15–21. Zbl0665.15006
  57. Yang Y. (1997): Robust System Design: Pole Assignment Approach, Ph.D. Thesis, Dept. Electr. Eng., University of Maryland at College Park, MD 20742, 1997. 
  58. Yang Y. and Tits A.L. (1995): Globally convergent algorithms for robust pole assignment by state feedback. — Tech. Rep., Dept. Electr. Eng. and Inst. Syst. Res., University of Maryland at College Park, MD 20742. 
  59. Yang Y. and Tits A.L. (1993): On robust pole assignment by state feedback. — Proc. Amer. Contr. Conf., San Francisco, pp.2765–2766. 
  60. Zhao Q. (1996): Measures for Least Change Secant Methods. — M.Sc. Thesis, Dept. Combinat. Optim., University of Waterloo, Ontario. 

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