Journal of Mechanical Science and Technology

The Korean Society of Mechanical Engineers (대한기계학회)
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Eigenstructure Assignment for Linear Systems with Probabilistic Uncertainties

  • Seo, Young-Bong (School of Mechanical Engineering and RIMT, Pusan National University) ;
  • Park, Jae-Weon (School of Mechanical Engineering and RIMT, Pusan National University) ;
  • Lee, Dal-Ho (Department of Electronics Engineering, Kyungwon University)
  • Published : 2004.06.01
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Abstract

In this paper, S (stochastic)-eigenvalue concept and its S-eigenvector for linear continuous-time systems with probabilistic uncertainties is proposed. The proposed concept is concerned with the perturbation of eigenvalues due to the probabilistic variable parameters in the dynamic model of a plant. S-eigenstructure assignment scheme via the Sylvester equation approach based on the S-eigenvalue concept is also proposed. The proposed design schemes are illustrated by numerical examples, and applied to the longitudinal dynamics of open-loop-unstable aircraft with possible uncertainties in aerodynamic and thrust effects as well as separate dynamic pressure. These results explicitly characterize how S-eigenvalues in the complex plane may impose stability on S-eigenstructure assignment.

Keywords

References

  1. Anderson, B. D. O. and Moore, J. B., 1971, Linear Optimal Control, NJ : Prentice-Hall
  2. Aoki, M., 1984, 'Control of Linear Discrete-Time Stochastic Dynamic Systems with Multiplicative Disturbances,' IEEE Transactions on Automatic Control, Vol. AC-20, pp. 388-392 https://doi.org/10.1109/TAC.1975.1100960
  3. Choi, J. W. and Seo, Y. B., 1999, 'LQR Design with Eigenstructure Assignment Capability,' IEEE Transactions on Aerospace and Electronic Systems, Vol. 35, No. 2, pp. 700-708 https://doi.org/10.1109/7.766949
  4. Dorato, P., 1987, 'A historical Review of Robust Control,' IEEE Control Systems Magazine, Vol. 7, pp. 44-47 https://doi.org/10.1109/MCS.1987.1105273
  5. Doyle, J. C., 1982, 'Analysis of Feedback Systems with Structured Uncertainties,' IEE Proc., Vol. 129, Pt. D., No. 6, pp. 242-250 https://doi.org/10.1049/ip-d.1982.0053
  6. Garg, S., 1991, 'Robust Eigenspace Assignment Using Singular Value Sensitivities,' Journal of Guidance, Control, and Dynamics, Vol. 14, No. 2, pp. 416-424 https://doi.org/10.2514/3.20654
  7. Fahmy, M. M. and Tantaway, H. S., 1984, 'Eigenstructure Assignment via Linear State-Feedback Control,' International Journal of Control, Vol. 40, No. 1, pp. 161-178 https://doi.org/10.1080/00207178408933265
  8. Horn, R. A. and Johnson, C. R., 1985, Matrix Analysis, Cambridge University Press
  9. Horowitz, I., 1982, 'Quantitative Feedback Theory,' IEE proc., Vol. 129, Pt. D., No. 6, pp. 215-226 https://doi.org/10.1049/ip-d.1982.0050
  10. John, S. B., 1999, Fundamentals of Linear State Space Systems, WCB/ McGraw-Hill, pp. 147-193
  11. Leon-Garcia, A., 1994, Probability and Random Processes for Electrical Engineering, Addison-Wesley Publishing Company, Inc.
  12. Lethomaki, N. A., Sandell, N. R. and Athans, M., 1981, 'Robustness Results in Linear Quadratic Gaussian Based Multivariable Control Designs,' IEEE Transactions on Automatic Control, Vol. AC-26, No. 1, pp. 75-92 https://doi.org/10.1109/TAC.1981.1102565
  13. Levinson, W. L., Kleinman, D. L. and Baron, S., 1968, 'A Model for Human Controller Remnant,' Bolt, Beranek and Newman, Inc., Tech. Rep. 1731
  14. Lin, C. F., 1994, Advanced Control Systems Design, Englewood Cliffs, NJ, Prentice Hall, Inc.
  15. Liu, G. P. and Patton, R. J., 1998, Eigenstructure Assignment for Control System Design, John Wiley & Sons
  16. Maybeck, P. S., 1979, Stochastic Models, Estimation, and Control, Vol. 1, Mathematics in Science and Engineering, Vol. 141, Academic Press, pp. 101-111
  17. Mohler, R. R. and Kolodziej, W. J., 1980, 'An Overview of Stochastic Bilinear Control Process,' IEEE Transactions on Systems, Man and Cybernetics, Vol. SMC-10, pp. 913-918
  18. Patel, R. V., Toda, M. and Sridhar, B., 1977, 'Robustness of Linear Quadratic State Feedback Designs in the Presence of System Uncertainty,' IEEE Transactions on Automatic Control, Vol. AC-22, pp. 945-949 https://doi.org/10.1109/TAC.1977.1101658
  19. Ray, L. R., 1992, 'Robust Linear-Optimal Control Laws for Active Suspension Systems,' Journal of Dynamic Systems, Measurement, and Control, Vol. 114, pp. 592-597 https://doi.org/10.1115/1.2897729
  20. Safonov, M. G. and Athans, M., 1977, 'Gains and Phase Margin for Multiloop LQG Regulators,' IEEE Transactions on Automatic Control, Vol. AC-22, pp. 173-179 https://doi.org/10.1109/TAC.1977.1101470
  21. Sandell, N. R. Jr., 1979, 'Recent Developments in the Robustness Theory of Multivariable Systems,' Office of Naval Research, Rep. No. ONR-CR215-271-1F
  22. Seo, Y. B. and Choi, J. W., 2003, Frequency Domain Properties of EALQR with Indefinite Weighting Matrix, KSME International Journal, Vol. 17, No. 3, pp. 329-335 https://doi.org/10.1007/BF02984358
  23. Sobel, K. M. and Lallman, F. J., 1989, 'Eigen-structure Assignment for the Control of Highly Augmented Aircraft,' Journal of Guidance, Control, and Dynamics, Vol. 12, No. 3, pp. 318-324 https://doi.org/10.2514/3.20411
  24. Sobel, K. M. and Shapiro, E. Y., 1985a, 'A Design Methodology for Pitch Pointing Flight Control Systems,' Journal of Guidance, Control and Dynamics, Vol. 8, No. 2, pp. 181-187 https://doi.org/10.2514/3.19957
  25. Sobel, K. M. and Shapiro, E. Y., 1985b, 'Eigenstructure Assignment for Design of Multimode Flight Control Systems,' IEEE Control Systems Magazine, pp. 9-15 https://doi.org/10.1109/MCS.1985.1104941
  26. Sobel, K. M., Shapiro, E. Y. and Andry, A. N., 1994, 'Eigenstructure Assignment,' International Journal of Control, Vol. 59, No. 1, pp. 13-37 https://doi.org/10.1080/00207179408923068
  27. Stengel, R. F., 1980, 'Some Effects of Parameter Variations on the Lateral-Directional Stability of Aircraft,' Journal of Guidance, Control and Dynamics, Vol. 3, No. 2, pp. 124-131 https://doi.org/10.2514/3.55959
  28. Stengel, R. F. and Ray, L. R., 1991, 'Stochastic Robustness of Linear Time-Invariant Control Systems,' IEEE Transactions on Automatic Control, Vol. 36, No. 1, pp. 82-87 https://doi.org/10.1109/9.62270
  29. Tahk, M. and Speyer, J. L., 1987, 'Modelling of Parameter Variations and Asymptotic LQG Synthesis,' IEEE Transactions on Automatic Control, Vol. AC-32, No. 9, pp. 793-801 https://doi.org/10.1109/TAC.1987.1104723
  30. Wingerden, A. J. M. and Dekoning, W. L., 1984, 'The Influence of Finite Word Length on Digital Optimal Control,' IEEE Transactions on Automatic Control, Vol. AC-29, pp. 385-391 https://doi.org/10.1109/TAC.1984.1103542
  31. Yaz, E., 1988, 'Deterministic and Stochastic Robustness Measures for Discrete Systems,' IEEE Transactions on Automatic Control, Vol. 33, No. 10, pp. 952-955 https://doi.org/10.1109/9.7253
  32. Yedavalli, R. K., Banda, S. S. and Ridgely, D. B., 1985, 'Time-Domain Stability Robustness Measures for Linear Regulators,' Journal of Guidance, Control, and Dynamics, Vol. 8, pp. 520-524 https://doi.org/10.2514/3.20014
  33. Yedvalli, R. K. and Liang, Z., 1986, 'Reduced Conservatism in Stability Robustness Bounds by State Transformation,' IEEE Transactions on Automatic Control, Vol. AC-31, No. 9, pp. 863-866 https://doi.org/10.1109/TAC.1986.1104408
  34. Zhou, K. and Khargonekar, P. P., 1987, 'Stability Robustness Bounds for Linear State-Space Models with Structured Uncertainty,' IEEE Transactions on Automatic Control, Vol. AC-32, pp. 621-623 https://doi.org/10.1109/TAC.1987.1104667