Journal of Institute of Control, Robotics and Systems (제어로봇시스템학회논문지)

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

DOI QR Code

Robust Optimal Bang-Bang Controller Using Lyapunov Robust Stability Condition

Lyapunov 강인 안정성 조건을 이용한 강인 최적 뱅뱅 제어기

  • Published : 2006.05.01
Download PDF
⟨ Previous Next ⟩

Abstract

There are mainly two types of bang-bang controllers for nominal linear time-invariant (LTI) system. Optimal bang-bang controller is designed based on optimal control theory and suboptimal bang-bang controller is obtained by using Lyapunov stability condition. In this paper, the suboptimal bang-bang control method is extended to LTI system involving both control input saturation and structured real parameter uncertainties by using Lyapunov robust stability condition. Two robust optimal bang-bang controllers are derived by minimizing the time derivative of Lyapunov function subjected to the limit of control input. The one is developed based on the classical quadratic stability(QS), and the other is developed based on the affine quadratic stability(AQS). And characteristics of the two controllers are compared. Especially, bounds of parameter uncertainties which theoretically guarantee robust stability of the two controllers are compared quantitatively for 1DOF vibrating system. Moreover, the validity of robust optimal bang-bang controller based on the AQS is shown through numerical simulations for this system.

Keywords

References

  1. D. S. Bernstein and A. N. Michel, 'A chronological biblography on saturating actuators,' International Journal of Robust and Nonlinear Control, vol. 5, pp. 375-380, 1995 https://doi.org/10.1002/rnc.4590050502
  2. W. M. Wonham and C. D. Johnson, 'Optimal bang-bang control with quadratic performance index,' Transactions on ASME, Ser. D, vol. 86, pp. 107-115, 1964
  3. B. Friendland, 'Limiting forms of optimum stochastic linear regulators,' Journal of Dynamic Systems Measurement Control, Transactions on ASME, Ser. G, vol. 93, no. 3, pp. 135-141, 1971
  4. L. Meirovitch, Dynamics and Control of Structures, Wiely, New York, 1990
  5. Z. Wu and T. T. Soong, 'Modified bang-bang control law for structural control implementation,' Journal Engineering Mechanics, ASCE, vol. 122, pp. 771-777, 1996 https://doi.org/10.1061/(ASCE)0733-9399(1996)122:8(771)
  6. P. P. Khargonekar, I. R. Petersen, and K. Zhou, 'Robust stabilization of uncertain linear systems : quadratic stabilizability and $H_{\infty}$ control theory,' IEEE Transactions on Automatic Control, vol. 35, no. 3, pp. 356-361, 1990 https://doi.org/10.1109/9.50357
  7. K. Wei, 'Quadratic stabilizability of linear systems with structural independent time-varying uncertainties,' IEEE Transactions on Automatic Control, vol. 35, no. 3, pp. 268-277, 1990 https://doi.org/10.1109/9.50337
  8. L. Xie, M. Fu, and C. E. de Souza, '$H_{\infty}$ control and quadratic stabilization of systems with parameter uncertainty via output feedback,' IEEE Transactions on Automatic Control, vol. 37, no. 8, pp. 268-277, 1992 https://doi.org/10.1109/9.121633
  9. S. S. L. Chang and T. K. C. Peng, 'Adaptive guaranteed cost control of systems with uncertain parameters,' IEEE Transactions on Automatic Control, vol. 17, pp. 474-483, 1972 https://doi.org/10.1109/TAC.1972.1100037
  10. B. N. Jain, 'Guaranteed error estimation in uncertain systems,' IEEE Transactions on Automatic Control, vol. 20, 1975 https://doi.org/10.1109/TAC.1975.1100926
  11. D. S. Bernstein, 'Robust stability and dynamic output-feedback stabilization : deterministic and stochastic perspectives,' IEEE Transactions on Automatic Control, vol. 32, pp. 1076-1084, 1987 https://doi.org/10.1109/TAC.1987.1104517
  12. O. I. Kosmidou and P. Bertrand, 'Robust-controller design for systems with large parameter variations,' International Journal on Control, vol. 45, pp. 927-938, 1987 https://doi.org/10.1080/00207178708933778
  13. D. S. Bernstein and W. M. Haddad, 'Robust stability and performance analysis for state space systems via quadratic Lyapunov bounds,' SIAM Journal on Matrix Analysis and Applications, vol. 11, pp. 239-271, 1990 https://doi.org/10.1137/0611017
  14. H. P. Horisberger and P. R. Belanger, 'Regulators for linear, time invariant plants with uncertain parameters,' IEEE Transactions on Automatic Control, vol. 21, pp. 705-708, 1976 https://doi.org/10.1109/TAC.1976.1101350
  15. D. S. Bernstein and S. L. Osburn, 'Guaranteed cost inequalities for robust stability and performance analysis,' International Journal of Robust and Nonlinear Control, vol. 12, pp. 1275-1297, 2002 https://doi.org/10.1002/rnc.691
  16. P. Gahinet and A. Nemirovski, The LMI Control Toolbox, The MathWorks Inc., 1995
  17. P. Gahinet, P. Apkarian, and M. Chilali, 'Affine parameter-dependent lyapunov functions and real parametric uncertainty,' IEEE Transactions on Automatic Control, vol. 41, no. 3, pp. 436-442, 1996 https://doi.org/10.1109/9.486646
  18. E. Feron, P. Apkarian, and P. Gahinet, 'Analysis and synthesis of robust control systems via parameter-dependent lyapunov functions,' IEEE Transactions on Automatic Control, vol. 41, no. 7, pp. 1041-1046, 1996 https://doi.org/10.1109/9.508913
  19. J. C. Geromel, M. C. de Oliveria, and L. Hsu, 'LMI characterization of structural and robust stability,' Linear Algebra and Its Application, vol. 285, no. 1-3, pp. 69-80, 1998 https://doi.org/10.1016/S0024-3795(98)10123-4
  20. D. Peaucelle, D. Arzelier, O. Bachelier, and J. Bemussou, 'A new robust D-stability condition for real convex polytopic uncertainty,' System & Control Letters, vol. 40, pp. 21-30, 2000 https://doi.org/10.1016/S0167-6911(99)00119-X
  21. A. Trofino and C. E. de Souza, 'Biquadratic stability of uncertain linear systems,' IEEE Transactions on Automatic Control, vol. 46, no. 8, pp. 1303-1307, 2001 https://doi.org/10.1109/9.940939