An SQP trust region method for solving the discrete-time linear quadratic control problem

El-Sayed M.E. Mostafa

International Journal of Applied Mathematics and Computer Science (2012)

  • Volume: 22, Issue: 2, page 353-363
  • ISSN: 1641-876X

Abstract

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In this paper, a sequential quadratic programming method combined with a trust region globalization strategy is analyzed and studied for solving a certain nonlinear constrained optimization problem with matrix variables. The optimization problem is derived from the infinite-horizon linear quadratic control problem for discrete-time systems when a complete set of state variables is not available. Moreover, a parametrization approach is introduced that does not require starting a feasible solution to initiate the proposed SQP trust region method. To demonstrate the effectiveness of the method, some numerical results are presented in detail.

How to cite

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El-Sayed 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 22.2 (2012): 353-363. <http://eudml.org/doc/208113>.

@article{El2012,
abstract = {In this paper, a sequential quadratic programming method combined with a trust region globalization strategy is analyzed and studied for solving a certain nonlinear constrained optimization problem with matrix variables. The optimization problem is derived from the infinite-horizon linear quadratic control problem for discrete-time systems when a complete set of state variables is not available. Moreover, a parametrization approach is introduced that does not require starting a feasible solution to initiate the proposed SQP trust region method. To demonstrate the effectiveness of the method, some numerical results are presented in detail.},
author = {El-Sayed M.E. Mostafa},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {output feedback control design; sequential quadratic programming; trust region method},
language = {eng},
number = {2},
pages = {353-363},
title = {An SQP trust region method for solving the discrete-time linear quadratic control problem},
url = {http://eudml.org/doc/208113},
volume = {22},
year = {2012},
}

TY - JOUR
AU - El-Sayed M.E. Mostafa
TI - An SQP trust region method for solving the discrete-time linear quadratic control problem
JO - International Journal of Applied Mathematics and Computer Science
PY - 2012
VL - 22
IS - 2
SP - 353
EP - 363
AB - In this paper, a sequential quadratic programming method combined with a trust region globalization strategy is analyzed and studied for solving a certain nonlinear constrained optimization problem with matrix variables. The optimization problem is derived from the infinite-horizon linear quadratic control problem for discrete-time systems when a complete set of state variables is not available. Moreover, a parametrization approach is introduced that does not require starting a feasible solution to initiate the proposed SQP trust region method. To demonstrate the effectiveness of the method, some numerical results are presented in detail.
LA - eng
KW - output feedback control design; sequential quadratic programming; trust region method
UR - http://eudml.org/doc/208113
ER -

References

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  1. Conn, A.R., Gould, N.I.M. and Toint, Ph.L. (2000). TrustRegion Methods, SIAM, Philadelphia, PA. 
  2. Garcia, G., Pradin, B. and Zeng, F. (2001). Stabilization of discrete-time linear systems by static output feedback, IEEE Transactions on Automatic Control 46(12): 1954-1958. Zbl1009.93066
  3. Kočvara M., Leibfritz, F., Stingl, M. and Henrion, D. (2005). A nonlinear SDP algorithm for static output feedback problems in COMPlib, Proceedings of the 16th IFAC World Congress on Automatic Control, Prague, Czech Republic, (on CDROM). 
  4. Lee, J.-W. and Khargonekar, P.P. (2007). Constrained infinitehorizon linear quadratic regulation of discrete-time systems, IEEE Transactions on Automatic Control 52(10): 1951-1958. 
  5. Leibfritz, F. (2004). COMPlib: COnstraint Matrixoptimization Problem library-A collection of test examples for nonlinear semi-definite programs, control system design and related problems, Technical report, http://www.complib.de/. 
  6. Leibfritz, F. and Mostafa, E.M.E. (2002). An interior point constrained trust region method for a special class of nonlinear semidefinite programming problems, SIAM Journal on Optimization 12(4): 1048-1074. Zbl1035.90102
  7. Leibfritz, F. and Mostafa, E.M.E. (2003). Trust region methods for solving the optimal output feedback design problem, International Journal of Control 76(5): 501-519. Zbl1040.93023
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  9. Mostafa, E.M.E. (2005a). A trust region method for solving the decentralized static output feedback design problem, Journal of Applied Mathematics & Computing 18(1-2): 1-23. Zbl1083.49024
  10. Mostafa, E.M.E. (2005b). An augmented Lagrangian SQP method for solving some special class of nonlinear semidefinite programming problems, Computational and Applied Mathematics 24(3): 461-486. Zbl1213.49044
  11. Mostafa, E.M.E. (2008). Computational design of optimal discrete-time output feedback controllers, Journal of the Operations Research Society of Japan 51(1): 15-28. Zbl1176.90575
  12. Mostafa, E.M.E. (2012). A conjugate gradient method for discrete-time output feedback control design, Journal of Computational Mathematics 30(3): 279-297. Zbl1265.93104
  13. Nocedal J. and Wright, S.J. (1999). Numerical Optimization, Springer, New York, NY. Zbl0930.65067
  14. Peres, P.L.D. and Geromel, J.C. (1993). H₂ control for discretetime systems optimality and robustness, Automatica 29(1): 225-228. Zbl0782.49022
  15. Sulikowski, B., Gałkowski, K., Rogers, E. and Owens, D.H. (2004). Output feedback control of discrete linear repetitive processes, Automatica 40(12): 2167-2173. Zbl1065.93019
  16. Syrmos, V.L., Abdallah, C.T., Dorato, P. and Grigoriadis, K. (1997). Static output feedback-A survey, Automatica 33(2): 125-137. Zbl0872.93036
  17. Varga, A. and Pieters, S. (1998). Gradient-based approach to solve optimal periodic output feedback control problems, Automatica 34(4): 477-481. Zbl0931.93043
  18. Zhai, G., Matsumoto, Y., Chen, X., Imae, J. and Kobayashi, T. (2005). Hybrid stabilization of discrete-time LTI systems with two quantized signals, International Journal of Applied Mathematics and Computer Science 15(4): 509-516. El-Sayed M.E. Mostafa received the B.Sc. degree in mathematics from Alexandria University, Egypt, in 1989, the M.Sc. degree in industrial mathematics from the University of Kaisersluatern, Germany, in 1994, and the Ph.D. degree in numerical optimization from Alexandria University in 2000. Since 2008 he has been working as an associate professor at the Department of Mathematics, Faculty of Science, Alexandria University. His research interests include numerical optimization and optimal control with engineering applications. Zbl1127.93355

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