Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2014 (2014), Article ID 683797, 14 pages
http://dx.doi.org/10.1155/2014/683797
Research Article

On the Solution of a Nonlinear Semidefinite Program Arising in Discrete-Time Feedback Control Design

Department of Mathematics, Faculty of Science, Alexandria University, Moharam Bey, Alexandria 21511, Egypt

Received 20 May 2013; Accepted 11 November 2013; Published 2 January 2014

Academic Editor: Jung-Fa Tsai

Copyright © 2014 El-Sayed M. E. Mostafa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. F. Jarre, “An interior-point method for non-convex semi-definite programs,” Optimization and Engineering, vol. 1, pp. 347–372, 2000. View at Publisher · View at Google Scholar
  2. F. Leibfritz and E. M. E. Mostafa, “An interior point constrained trust region method for a special class of nonlinear semidefinite programming problems,” SIAM Journal on Optimization, vol. 12, no. 4, pp. 1048–1074, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. M. Kočvara, F. Leibfritz, M. Stingl, and D. Henrion, “A nonlinear sdp algorithm for static output feedback problems in COMPleib,” in Proceedings of the 16th Triennial World Congress of International Federation of Automatic Control (IFAC '05), pp. 1055–1060, July 2005. View at Scopus
  4. D. Sun, J. Sun, and L. Zhang, “The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming,” Mathematical Programming, vol. 114, no. 2, pp. 349–391, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. R. Correa and H. C. Ramirez, “A global algorithm for nonlinear semidefinite programming,” SIAM Journal on Optimization, vol. 15, no. 1, pp. 303–318, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. H. Yamashita and H. Yabe, “Local and superlinear convergence of a primal-dual interior point method for nonlinear semidefinite programming,” Mathematical Programming, vol. 132, no. 1-2, pp. 1–30, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. R. W. Freund, F. Jarre, and C. H. Vogelbusch, “Nonlinear semidefinite programming: sensitivity, convergence, and an application in passive reduced-order modeling,” Mathematical Programming, vol. 109, no. 2-3, pp. 581–611, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. F. Leibfritz, “COMPlib: COnstraint Matrix-optimization Problem library-a collection of test examples for nonlinear semi-definite programs, control system design and related problems,” Tech. Rep., 2004, http://www.complib.de/. View at Google Scholar
  9. P. M. Mäkilä and H. T. Toivonen, “Computational methods for parametric LQ problems—a survey,” IEEE Transactions on Automatic Control, vol. 32, no. 8, pp. 658–671, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. T. Rautert and E. W. Sachs, “Computational design of optimal output feedback controllers,” SIAM Journal on Optimization, vol. 7, no. 3, pp. 837–852, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. P. M. Mäkilä, “Linear quadratic control revisited,” Automatica, vol. 36, no. 1, pp. 83–89, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. P. Apkarian, H. D. Tuan, and J. Bernussou, “Continuous-time analysis, eigenstructure assignment, and H2 synthesis with enhanced linear matrix inequalities (LMI) characterizations,” IEEE Transactions on Automatic Control, vol. 46, no. 12, pp. 1941–1946, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  13. F. Leibfritz and E. M. E. Mostafa, “Trust region methods for solving the optimal output feedback design problem,” International Journal of Control, vol. 76, no. 5, pp. 501–519, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. E. M. E. Mostafa, “A trust region method for solving the decentralized static output feedback design problem,” Journal of Applied Mathematics & Computing, vol. 18, no. 1-2, pp. 1–23, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. E. M. E. Mostafa, “An augmented Lagrangian SQP method for solving some special class of nonlinear semi-definite programming problems,” Computational & Applied Mathematics, vol. 24, no. 3, pp. 461–486, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. E. M. E. Mostafa, “An SQP trust region method for solving the discrete-time linear quadratic control problem,” International Journal of Applied Mathematics and Computer Science, vol. 22, no. 2, pp. 353–363, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  17. J. Nocedal and S. J. Wright, Numerical Optimization, Springer Series in Operations Research, Springer, New York, NY, USA, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  18. V. L. Syrmos, C. T. Abdallah, P. Dorato, and K. Grigoriadis, “Static output feedback-a survey,” Automatica, vol. 33, pp. 125–137, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. G. Garcia, B. Pradin, and F. Zeng, “Stabilization of discrete time linear systems by static output feedback,” IEEE Transactions on Automatic Control, vol. 46, no. 12, pp. 1954–1958, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. J.-W. Lee and P. P. Khargonekar, “Constrained infinite-horizon linear quadratic regulation of discrete-time systems,” IEEE Transactions on Automatic Control, vol. 52, no. 10, pp. 1951–1958, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  21. E. M. E. Mostafa, “Computational design of optimal discrete-time output feedback controllers,” Journal of the Operations Research Society of Japan, vol. 51, no. 1, pp. 15–28, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. E. M. E. Mostafa, “A conjugate gradient method for discrete-time output feedback control design,” Journal of Computational Mathematics, vol. 30, no. 3, pp. 279–297, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. B. Sulikowski, K. Galkowski, E. Rogers, and D. H. Owens, “Output feedback control of discrete linear repetitive processes,” Automatica, vol. 40, no. 12, pp. 2167–2173, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. E. D. Sontag, Mathematical Control Theory, vol. 6, Springer, New York, NY, USA, 2nd edition, 1998. View at MathSciNet
  25. F. S. Kupfer, Reduced SQP in Hilbert space with applications to optimal control, [Ph.D. dissertation], FB IV-Mathematik, Universitat Trier, Trier, Germany, 1992.
  26. H. Wang and S. Daley, “A fault detection method for unknown systems with unknown input and its application to hydraulic turbine monitoring,” International Journal of Control, vol. 57, pp. 247–260, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH