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). 1. Introduction In this paper, the following nonlinear semidefinite programming (NSDP) problem is considered: where , , are sufficiently smooth matrix functions, where means that is positive definite. This problem is assumed to be nonlinear and generally nonconvex. The origin of the considered NSDP problem is the static output feedback (SOF) design problem in which the unknown is partitioned as , where represents the state variable and represents the control variable. In the last decade NSDP has attracted the attention of many authors in the optimization community. For instance, Jarre [1] introduced an interior-point method for nonconvex semi-definite programs. Leibfritz and Mostafa [2] proposed an interior-point trust region method for a special class of NSDP problems resulting from the continuous-time SOF problem. Ko？vara et al. [3] considered an augmented Lagrangian method for a similar NSDP problem. Sun et al. [4] investigated the rate of convergence of the augmented Lagrangian approach for NSDP. Correa and Ramirez [5] proposed sequential semi-definite programming method for nonlinear semi-definite programming. Yamashita and Yabe [6] study local and superlinear convergence of a primal-dual interior point method. Freund et al. [7] proposed a sequential semidefinite programming approach for a nonlinear program with nonlinear semidefinite constraints. The design problem of optimal output feedback controllers is one of the most studied problems over the last four decades in the community of systems and control. Clearly, this is due to the existence of numerous practical applications in system and control engineering, finance, and statistics. The benchmark collection [8], for instance, contains over 160 applications from system and control engineering only. M？kil？ and Toivonen in the survey [9] summarized several special purpose algorithms such as the Levine-Athans method, the dual Levine-Athans method, and the descent Anderson-Moore method as well as Newton’s method for finding the local solution of a matrix optimization problem that corresponds to the
References
[1]
F. Jarre, “An interior-point method for non-convex semi-definite programs,” Optimization and Engineering, vol. 1, pp. 347–372, 2000.
[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.
[3]
M. Ko？vara, 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.
[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.
[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.
[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.
[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.
[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/.
[9]
P. M. M？kil？ 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.
[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.
[11]
P. M. M？kil？, “Linear quadratic control revisited,” Automatica, vol. 36, no. 1, pp. 83–89, 2000.
[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.
[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.
[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.
[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.
[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.
[17]
J. Nocedal and S. J. Wright, Numerical Optimization, Springer Series in Operations Research, Springer, New York, NY, USA, 1999.
[18]
V. L. Syrmos, C. T. Abdallah, P. Dorato, and K. Grigoriadis, “Static output feedback-a survey,” Automatica, vol. 33, pp. 125–137, 1997.
[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.
[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.
[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.
[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.
[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.
[24]
E. D. Sontag, Mathematical Control Theory, vol. 6, Springer, New York, NY, USA, 2nd edition, 1998.
[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.
