## Journal of Applied Mathematics

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

El-Sayed M. E. Mostafa

#### Abstract

A sequential quadratic programming method with line search is analyzed and studied for finding the local solution of a nonlinear semidefinite programming problem resulting from the discrete-time output feedback problem. The method requires an initial feasible point with respect to two positive definite constraints. By parameterizing the optimization problem we ease that requirement. The method is tested numerically on several test problems chosen from the benchmark collection (Leibfritz, 2004).

#### Article information

Source
J. Appl. Math., Volume 2014 (2014), Article ID 683797, 14 pages.

Dates
First available in Project Euclid: 26 March 2014

https://projecteuclid.org/euclid.jam/1395855266

Digital Object Identifier
doi:10.1155/2014/683797

Mathematical Reviews number (MathSciNet)
MR3166779

Zentralblatt MATH identifier
07010713

#### Citation

Mostafa, El-Sayed M. E. On the Solution of a Nonlinear Semidefinite Program Arising in Discrete-Time Feedback Control Design. J. Appl. Math. 2014 (2014), Article ID 683797, 14 pages. doi:10.1155/2014/683797. https://projecteuclid.org/euclid.jam/1395855266

#### References

• F. Jarre, “An interior-point method for non-convex semi-definite programs,” Optimization and Engineering, vol. 1, pp. 347–372, 2000.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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/.
• 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.
• 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.
• P. M. Mäkilä, “Linear quadratic control revisited,” Automatica, vol. 36, no. 1, pp. 83–89, 2000.
• P. Apkarian, H. D. Tuan, and J. Bernussou, “Continuous-time analysis, eigenstructure assignment, and H$_{2}$ synthesis with enhanced linear matrix inequalities (LMI) characterizations,” IEEE Transactions on Automatic Control, vol. 46, no. 12, pp. 1941–1946, 2001.
• 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.
• E. M. E. Mostafa, “A trust region method for solving the decen-tralized static output feedback design problem,” Journal of Applied Mathematics & Computing, vol. 18, no. 1-2, pp. 1–23, 2005.
• 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.
• 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.
• J. Nocedal and S. J. Wright, Numerical Optimization, Springer Series in Operations Research, Springer, New York, NY, USA, 1999.
• V. L. Syrmos, C. T. Abdallah, P. Dorato, and K. Grigoriadis, “Static output feedback-a survey,” Automatica, vol. 33, pp. 125–137, 1997.
• 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.
• 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.
• 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.
• 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.
• 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.
• E. D. Sontag, Mathematical Control Theory, vol. 6, Springer, New York, NY, USA, 2nd edition, 1998.
• F. S. Kupfer, Reduced SQP in Hilbert space with applications to optimal control, [Ph.D. dissertation], FB IV-Mathematik, Universitat Trier, Trier, Germany, 1992.
• 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. \endinput